Affine differential geometry and smoothness maximization as tools for identifying geometric movement primitives

Abstract

Neuroscientific studies of drawing-like movements usually analyze neural representation of either geometric (e.g., direction, shape) or temporal (e.g., speed) parameters of trajectories rather than trajectory's representation as a whole. This work is about identifying geometric building blocks of movements by unifying different empirically supported mathematical descriptions that characterize relationship between geometric and temporal aspects of biological motion. Movement primitives supposedly facilitate the efficiency of movements' representation in the brain and comply with such criteria for biological movements as kinematic smoothness and geometric constraint. The minimum-jerk model formalizes criterion for trajectories' maximal smoothness of order 3. I derive a class of differential equations obeyed by movement paths whose nth-order maximally smooth trajectories accumulate path measurement with constant rate. Constant rate of accumulating equi-affine arc complies with the 2/3 power-law model. Candidate primitive shapes identified as equations' solutions for arcs in different geometries in plane and in space are presented. Connection between geometric invariance, motion smoothness, compositionality and performance of the compromised motor control system is proposed within single invariance-smoothness framework. The derived class of differential equations is a novel tool for discovering candidates for geometric movement primitives.

Discovery and recognition of motion primitives in human activities

“…Likewise daily activities such as cleaning, or cooking, or washing the dishes or preparing the table require precise motion sequences to accomplish the task. Indeed, the compositional nature of human February 5, 2019 1/39 activities, under body and kinematics constraints, has attracted the interest of many research areas such as in computer vision [2,3], in neurophysiology [4,5], in sports and rehabilitation [6], and in biomechanics [7] and in robotics [8,9,10]. The goal of this work is to automatically discover the start and end points where primitives of 6 identified body parts occur throughout the course of an activity, and recognize each of the occurred primitives.…”

“…activities, under body and kinematics constraints, has attracted the interest of many research areas such as in computer vision [2,3], in neurophysiology [4,5], in sports and rehabilitation [6], and in biomechanics [7] and in robotics [8,9,10]. The goal of this work is to automatically discover the start and end points where primitives of 6 identified body parts occur throughout the course of an activity, and recognize each of the occurred primitives.…”

“…tom -rig ht (4) Tu rn lef t(5) Ro ll lef t (6 ) R ol l rig ht(7 ) T ur n bo tto m -l ef t(8 ) T u rn to p -r ig h t(9 ) T u rn to p -l e ft(1 0) B e n d f o r w a r d ( 1 ) R a is e to r s o ( 2 ) B e n d l e f t ( 3 ) B e n d r i g h t ( 4 ) R o ll le ft (5 ) R o ll ri g h t (u l d e r e x t r o t a n d e l b o w e x t e n ( 5 ) S h o u ld e r in t r o t a n d e lb o w e x te n ( 6 ) S h o u ld e r e x t ro t a n d e lb o w fl e x (7 ) S h o u ld e r in t ro t a n d e lb o w fl e x (8 ) S h o u ld er ad d u ct io n (9 ) S ho ul de r ci rc li ng (1 0) Sh ou ld er ex te rn al ro t (1 1) Sh ou ld er in te rn al ro t (1 2) El bo w low er me dia l fle x (13 ) Elb ow upp er me dia l flex (14 ) Elbo w low er med ial exte n (15) Elbow upper media l exten (16) ulde r ext rot and elbo w exte n (5) Sho uld er int rot and elb ow ext en (6) Sh ou lde r ex t rot an d elb ow fle x (7) Sh ou ld er in t ro t an d el bo w fle x (8 ) Sh ou ld er ad du ct io n (9 ) S ho ul de r ci rc li ng (1 0) S h o u ld er ex te rn al ro t (1 1 ) S h o u ld e r in te rn a l ro t (1 2 ) E lb o w lo w e r m e d ia l fl e x (1 3 ) E lb o w u p p e r m e d ia l f le x ( 1 4 ) E l b o w l o w e r m e d i a l e x t e n ( 1 5 ) E l b o w u p p e r m e d i a l e x t e n ( e x t e r n a l r o t ( 8 ) K n e e lo w e r m e d ia l f le x ( 9 ) K n e e lo w e r m e d ia l e x te n ( 1 0 ) H ip fl e x (1 )H ip ex te n (2 ) H ip ab du ct io n (3 ) H ip ad du ct io n (4 ) Kn ee fle x (5 ) Kn ee ext en(6) Leg inte rna l rot(7) Leg exte rnal rot (8) Knee lower medial flex (9) Knee lower medial exten(10) Diagram showing the motion primitives of each group. Abbreviation ext stands for external, int for internal, rot for rotation, exten for extension, and flex for flexion.…”