Bipedal Walkers: From Three to Bipedal Walkers: From Three to Two - PowerPoint PPT Presentation

Bipedal Walkers: From Three to Bipedal Walkers: From Three to Two Dimensions via Lagrangian Lagrangian Two Dimensions via Reduction Reduction Robert D Gregg IV Undergraduate Researcher, University of California, Berkeley

Problem of 3D Walkers Problem of 3D Walkers 1.0 Background: Problem of 3D Bipedal Walkers 1.0 Background: Problem of 3D Bipedal Walkers 1.1 Analysis of 2D Walkers 1.1 Analysis of 2D Walkers 1.2 Application: Simple Compass- -Gait Biped Gait Biped 1.2 Application: Simple Compass 1.3 Scaling Complexity from 2D to 3D 1.3 Scaling Complexity from 2D to 3D 2.0 Hybrid Reduction from 3D to 2D 2.0 Hybrid Reduction from 3D to 2D 2.1 Hybridization of Robot Motion 2.1 Hybridization of Robot Motion 2.2 Discrete Foot Impact 2.2 Discrete Foot Impact 2.3 Lagrangian Lagrangian Continuous Dynamics Continuous Dynamics 2.3 2.4 Dependency Simplification of Lagrangian 2.4 Dependency Simplification of Lagrangian 2.5 Routhian Routhian Reduction Reduction 2.5 3.0 Results 3.0 Results 3.1 Reduced Model 3.1 Reduced Model 3.2 Equations of Motion (2D) 3.2 Equations of Motion (2D) 3.3 Hypothesis of 3D Motion 3.3 Hypothesis of 3D Motion 4.0 Final Thoughts 4.0 Final Thoughts

Analysis of 2D Walkers Analysis of 2D Walkers � Many techniques have already been established for analyzing Many techniques have already been established for analyzing � two dimensional bipedal walkers two dimensional bipedal walkers � Finding stable walking cycles Finding stable walking cycles � o Dynamics described by non Dynamics described by non- -linear linear ODEs ODEs o o No straightforward No straightforward backsolving backsolving method to find initial states method to find initial states o o Solution: Numerical analysis using methods of o Solution: Numerical analysis using methods of Poincar Poincaré é Search feasible phase space for initial states that result � Search feasible phase space for initial states that result � in asymptotically stable cycles in asymptotically stable cycles . Poincaré é Poincar Map Discrete Transition Continuous Map Dynamics x

Compass- -Gait Bipedal Walker (2D) Gait Bipedal Walker (2D) Compass y M 2 α m b m Θ ns l = a + b Θ s a Non-stance/swing leg Stance leg (pivot) x � Four state dependencies: Four state dependencies: Θ non-stance , Θ stance , and time-derivatives �

Compass- -Gait Bipedal Walker (3D) Gait Bipedal Walker (3D) Compass z M 2 α m b m Θ ns Θ s l = a + b a y Φ ns Φ s x � Eight state dependencies: Eight state dependencies: Θ non-stance , Θ stance , , Φ non-stance , Φ stance , � and time-derivatives

Scaling Complexity Scaling Complexity � Increasing the model Increasing the model’ ’s dimensions from two to three s dimensions from two to three � results in a two- -fold increase of state dependency fold increase of state dependency results in a two � Thus, in three dimensions, numerical analysis Thus, in three dimensions, numerical analysis � requires a phase space search of eight eight dimensions dimensions requires a phase space search of � Analysis is computably impractical! Analysis is computably impractical! � Solution : Hybrid Reduction on the 3D Model : Hybrid Reduction on the 3D Model � Solution �

Hybrid Reduction from 3D to 2D Hybrid Reduction from 3D to 2D 1.0 Background: Problem of 3D Bipedal Walkers 1.0 Background: Problem of 3D Bipedal Walkers 1.1 Analysis of 2D Walkers 1.1 Analysis of 2D Walkers 1.2 Application: Simple Compass- -Gait Biped Gait Biped 1.2 Application: Simple Compass 1.3 Scaling Complexity from 2D to 3D 1.3 Scaling Complexity from 2D to 3D 2.0 2.0 Hybrid Reduction from 3D to 2D Hybrid Reduction from 3D to 2D 2.1 Hybridization of Robot Motion 2.1 Hybridization of Robot Motion 2.2 Discrete Foot Impact Transition 2.2 Discrete Foot Impact Transition 2.3 Lagrangian Lagrangian Continuous Dynamics Continuous Dynamics 2.3 2.4 Dependency Simplification of Lagrangian Lagrangian 2.4 Dependency Simplification of 2.4.1 Fixing inner angle 2 γ 2.4.1 Fixing inner angle 2 γ 2.4.2 Limit as M/m approaches infinity 2.4.2 Limit as M/m approaches infinity 2.4.3 Fixing Φ s = Φ ns (x x- -y y plane) plane) 2.4.3 Fixing Φ s = Φ ns ( 2.5 2.5 Routhian Routhian Reduction Reduction 3.0 Results 3.0 Results 3.1 Reduced Model 3.1 Reduced Model 3.2 Equations of Motion (2D) 3.2 Equations of Motion (2D) 3.3 Hypothesis of 3D Motion 3.3 Hypothesis of 3D Motion 4.0 4.0 Final Thoughts Final Thoughts

Process of Reduction (General) Process of Reduction (General) Robot Structure Hybridization Lagrangian Formulation of Discrete Impact Continuous Dynamics Transition Map Dependency Simplification Hybrid Reduction

Hybridization Hybridization � System’s single-support phase guided by differential equations { Foot Impact } (continuous dynamics) Transition Map T � Swing leg’s impact with ground considered a reset Single-Support transition for hybrid Phase Dynamics: system (discrete event) q ( t ) = f ( q ( t )) ˙

Discrete Foot Impact Discrete Foot Impact � Impact Equations (swing leg impact on ground): Impact Equations (swing leg impact on ground): � � Angle positions preserved Angle positions preserved � � Discontinuity in angle velocity (different ways of modeling, Discontinuity in angle velocity (different ways of modeling, � Grizzle v Goswami Grizzle v Goswami) ) Swing Leg Impact γ � Transition Map (hybrid system reset) Transition Map (hybrid system reset) � � Swing leg becomes stance leg: angle positions swap Swing leg becomes stance leg: angle positions swap � Angle velocities: Θ ` + = H( γ ) Θ ` – � Angle velocities: – �

Lagrangian Formulation Formulation Lagrangian � The The Lagrangian Lagrangian formulation accounts for all energy in formulation accounts for all energy in � the system the system � Lagrangian Lagrangian = Kinetic Energy = Kinetic Energy – – Potential Energy Potential Energy � L = K – – V V L = K ½ Θ ’ T M( Θ ) Θ ’ – L = ½ – ∫ q( Θ ) L = � Derive the continuous Equations of Motion (passive): Derive the continuous Equations of Motion (passive): � M( Θ ) Θ ’’ + F + F( Θ , Θ ’) Θ ’ + q( Θ ) = 0 where Θ = [ Θ ns, Θ s, Φ ns, Φ s] T � M and F are 4x4 matrices and q is a 4x1 vector � Pages and pages of matrix entries!

Dependency Simplification Dependency Simplification � Goal is to find cyclic variables in Goal is to find cyclic variables in Lagrangian Lagrangian � Strategies: Strategies: � Fixing inner angle 2 Fixing inner angle 2 γ γ => No cyclic variables => No cyclic variables � � Limit as M/m approaches infinity => No cyclic Limit as M/m approaches infinity => No cyclic � � Limit as b/a approaches infinity => No cyclic Limit as b/a approaches infinity => No cyclic � � Fixing Fixing Φ Φ s[t s[t] = ] = Φ Φ ns[t ns[t] ( ] (x x- -y y plane) plane) � � Two Two cyclic variables: cyclic variables: Φ ns[t] and Φ s[t] � M[ Θ ] = M1 M2

Routhian Reduction Reduction Routhian � Φ ns[t] and Φ s[t] independent � M 2 ( Θ ) Φ ` = c (Routhian constant) where Φ ` = [ Φ `ns, Φ `s ] T , Θ = [ Θ ns, Θ s ] T , c = [ c 1 , c 2 ] T and Φ `ns[t] = Φ `s[t] Augmented Term � Solve: Φ ` = c/m( Θ ) Routhian = [ L( = [ L( Θ , , Θ `, Φ `) – – c Φ ` ] Φ `=c/m( Θ ) � Routhian � ½ Θ ` T M 1 ( Θ ) Θ ` – R = ½ – ∫ q( Θ ) – – ½ ½ c 2 /m( Θ ) R =

Results of Reduction Results of Reduction 1.0 1.0 Background: Problem of 3D Bipedal Walkers Background: Problem of 3D Bipedal Walkers 1.1 Analysis of 2D Walkers 1.1 Analysis of 2D Walkers 1.2 Application: Simple Compass- -Gait Biped Gait Biped 1.2 Application: Simple Compass 1.3 Scaling Complexity from 2D to 3D 1.3 Scaling Complexity from 2D to 3D 2.0 Hybrid Reduction from 3D to 2D 2.0 Hybrid Reduction from 3D to 2D 2.1 Hybridization of Robot Motion 2.1 Hybridization of Robot Motion 2.2 Discrete Foot Impact Transition 2.2 Discrete Foot Impact Transition 2.3 Lagrangian Lagrangian Continuous Dynamics Continuous Dynamics 2.3 2.4 Dependency Simplification of Lagrangian 2.4 Dependency Simplification of Lagrangian 2.5 Routhian Routhian Reduction Reduction 2.5 3.0 Results Results 3.0 3.1 Reduced Model 3.1 Reduced Model 3.2 Equations of Motion (2D) 3.2 Equations of Motion (2D) 3.3 Hypothesis of 3D Motion 3.3 Hypothesis of 3D Motion 4.0 Final Thoughts 4.0 Final Thoughts

Reduced Model Reduced Model � Continous Equations of Motion (passive): M( Θ ) Θ ’’ + F( + F( Θ , Θ ’) ) Θ ’ + q( + q( Θ ) + aug aug c ( Θ ) = 0 0 M( Θ ) Θ ’’ Θ , Θ ’ Θ ’ Θ ) + c ( Θ ) = where Θ = [ Θ ns, Θ s ] T Augmented Potential Original 2D Model Conclusions: � M and F are 2x2 matrices; q and aug are 2x1 vectors � The reduced model (now 2D) is equivalent to the original 2D model with an augmented term � Matrices M and F and vector q remain the same; overall potential term is modified. � Additional constant c (if zero => original 2D model) � Uniqueness: Trivial to bring back to unique 3D model

Normalized Eqns Eqns of Motion (2D) of Motion (2D) Normalized M( Θ ) = F( Θ , Θ `) = q( Θ ) = -aug c ( Θ ) =