Safe Execution of Bipedal Walking Tasks from Biomechanical Principles Andreas Hofmann and Brian Williams

Introduction

Problem: For agile, underactuated systems, can’t ignore dynamics

Introduction Problem: For agile, underactuated systems, can’t ignore dynamics

Introduction Problem: For agile, underactuated systems, can’t ignore dynamics Problem: No notion of task plan, little flexibility to disturbances

Introduction – Problem Addressed Gap: Large class of problems that require -ability to execute task-level plans -ability to deal with disturbances -taking into account dynamic limitations; understanding relationship between acceleration limits, and time needed to achieve state-space goals

Challenging case – bipedal walking Walk from location A to B in specified time Observe foot placement restrictions imposed by terrain

Challenging case – bipedal walking Walk from location A to B in specified time Observe foot placement restrictions imposed by terrain

Challenging case – Bipedal Machines Walk from location A to B in specified time

Challenging case – Bipedal Machines Walk from location A to B in specified time Should not fall, even if disturbed

Challenging case – Bipedal Machines Should not fall, even if disturbed

Challenging case – Bipedal Machines Should not fall, even on shaky ground

Challenging case – Bipedal Machines Should not fall, even on shaky ground

Challenging case – Bipedal Machines Should not fall, even on shaky ground But there are limits!

Approach – walking task spec Qualitative State Plan

Computing torques to achieve a particular state goal is challenging

Hybrid executive and multivariable controller

Hybrid executive coordinates controllers to sequence plant through poses in qualitative state plan

Multivariable controller makes state plan quantities, like CM, directly controllable allows hybrid executive to control CM by adjusting linear gain parameters

Innovations Requirement: Stable walking

Innovations Requirement: Stable walking Previous Approaches

Innovations Requirement: Stable walking Previous Approaches

Innovations Requirement: Stable walking How to get to the right place at the right time? What if terrain requires irregular foot placement? Previous Approaches

Innovations Requirement: Stable walking How to get to the right place at the right time? What if terrain requires irregular foot placement? Previous Approaches Innovation Execute a plan

Innovations Previous Approaches Detailed actuated trajectory spec. Requirement: ability to execute task-level plans How should walking plans be expressed? What are the requirements for successful plan execution?

Innovations Previous ApproachesInnovation Detailed actuated trajectory spec. Qualitative state trajectory spec. Requirement: ability to execute task-level plans How should walking plans be expressed? What are the requirements for successful plan execution?

Innovations Previous ApproachesInnovation Detailed actuated trajectory spec. Qualitative control plan Requirement: ability to execute task-level plans How should walking plans be expressed? What are the requirements for successful plan execution?

Innovations Requirement: ability to deal with disturbances What balance strategies can bipeds (like humans) use?

Innovations Requirement: ability to deal with disturbances What balance strategies can bipeds (like humans) use? Previous Approaches Uses primarily ankle torque strategy

Innovations Requirement: ability to deal with disturbances What balance strategies can bipeds (like humans) use? Previous Approaches Uses primarily ankle torque strategy Use three balance strategies Innovation

Humans use Three Balance Strategies Stance ankle torque Stepping Movement of non-contact segments

Innovations Requirement: account for dynamic limitations What is the relationship between acceleration limits, and timing needed to achieve state-space goals?

Innovations Requirement: account for dynamic limitations What is the relationship between acceleration limits, and timing needed to achieve state-space goals? Previous Approach – exploits waits [Morris, 2001]

Innovations Requirement: account for dynamic limitations What is the relationship between acceleration limits, and timing needed to achieve state-space goals? Previous Approach – exploits waits [Morris, 2001] Innovation Underactuated system - no equilibrium point (no ability to wait)

Problem Solution Take state plan and plant state as input Generate plant control input that causes plant state to evolve in accordance with the state plan specification.

Multivariable controller makes CM directly controllable

Multivariable Controller Requirements Want to specify coarse setpoint –Forward CM setpoint = 0 –Lateral CM setpoint = 0 Controller should figure out detailed joint trajectories

Hybrid executive decides CM setpoints, control gains adjusts kp, kd gains of SISO abstraction

Hybrid Executive Requirements Multivariable controller accepts single setpoint

Hybrid Executive Requirements Multivariable controller accepts single setpoint Can’t, by itself, sequence through multiple setpoints Need hybrid executive for that

At start of control epoch, hybrid exec. sets controller gains

Hybrid Executive guides each variable to its goal

Hybrid Executive transitions to next epoch when goal for each variable is achieved

What if there is a disturbance? trip recovery

Disturbances and Controllability How can disturbances be handled? Given some bound on disturbances, is it possible to guarantee successful execution of a plan? Dispatchers for discrete systems

Disturbances and Controllability How can disturbances be handled? Given some bound on disturbances, is it possible to guarantee successful execution of a plan? Dispatchers for discrete systems Guarantee successful execution Even with temporal uncertainty If uncertainty is bounded, [Morris, 2001]

Controllability for Hybrid Systems Executive guides variables to goal regions, but what should these regions be? Previous approaches [Pratt, et. al 1996] determine regions manually Can regions be computed automatically? –based on relation between regions, time, and controllability limits?

Plan compiler computes limits Computes spatial and temporal regions for all activities

Plan compiler synthesizes controllers Control info expressed as ranges on SISO parameters

Plan Compiler Generate qualitative control plan from state plan: Compute initial and goal regions for each activity Compute duration range for each activity Compute control parameter ranges Formulate as Nonlinear Program, and solve by SQP

How does the plan compiler compute region limits, synthesize controllers? Initial and goal regions

How does the plan compiler compute region limits, synthesize controllers? Initial and goal regions Want to maximize controllable time range in goal Given start anywhere in init region, what are lb, ub on this time?

How does the plan compiler compute region limits, synthesize controllers? Lb – fastest trajectory from slowest start Worst-case (slowest) start is point B

How does the plan compiler compute region limits, synthesize controllers? Lb – fastest trajectory from slowest start Worst-case (slowest) start is point B Best-case (fastest) finish is point D

How does the plan compiler compute region limits, synthesize controllers? Consider single acceleration spike as control input Spike occurs at beginning

How does the plan compiler compute region limits, synthesize controllers? Consider single acceleration spike as control input Spike occurs at beginning If spike has the right size, results in GFT (Guaranteed Fastest Trajectory)

How does the plan compiler compute region limits, synthesize controllers? Ub – slowest trajectory from fastest start Worst-case (fastest) start is point A Best-case (slowest) finish is point C

How does the plan compiler compute region limits, synthesize controllers? Spike of right size at end results in GST (Guaranteed Slowest Trajectory)

Existence of controllable temporal range in goal If t(GFT)<t(GST) then presence of trajectory in goal pos./vel. region can be guaranteed for any time [t(GFT), t(GST)] –By adjusting spike

GFT, GST with linear control law Adjust control law parameters to get GFT, GST A – max pos., vel. (fastest start) B – min pos., vel. (slowest start) C – max pos., min vel. (slowest finish) D – min pos., max vel. (fastest finish) Assume monotonic velocity Maximize controllable temporal range, initial region size Subject to limits on control inputs

Controllable Regions for CM 1 Lateral Forward 2 3 4

Discussion Hybrid executive –From qualitative state plan, automatically synthesizes controllers Computes dispatcher regions and gain ranges –Successful, stable execution achieved by getting key variables into right region at right time Provides significant flexibility in how they actually get there –Relies on SISO decoupling, linearization provided by multivariable controller

Conclusion Robustness achieved through integration of three balance control strategies Robust plan execution achieved for hybrid system by extending techniques used for discrete systems Efficiency of execution achieved through compilation of plan into dispatchable form

Addendum

Trade-off Between Region Size and Temporal Range t(GFT) = t(GST) GFT in red, GST in blue, Nom in green

Trade-off Between Region Size and Temporal Range t(GFT) = t(GST) GFT in red, GST in blue, Nom in green t(GFT) > t(GST) Some uncertainty in duration

Strong and Dynamic Hybrid Controllability