Safe Execution of Bipedal Walking Tasks from Biomechanical Principles

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Safe Execution of Bipedal Walking Tasks from
Biomechanical Principles

• Andreas Hofmann and Brian Williams

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Introduction
3
Introduction

• Problem For agile, underactuated systems, cant
  ignore dynamics

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Introduction

• Problem For agile, underactuated systems, cant
  ignore dynamics

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Introduction

• Problem For agile, underactuated systems, cant
  ignore dynamics

Problem No notion of task plan, little
flexibility to disturbances
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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

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Challenging case bipedal walking

• Walk from location A to B in specified time
• Observe foot placement restrictions imposed by
  terrain

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Challenging case bipedal walking

• Walk from location A to B in specified time
• Observe foot placement restrictions imposed by
  terrain

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Challenging case Bipedal Machines

• Walk from location A to B in specified time

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Challenging case Bipedal Machines

• Walk from location A to B in specified time
• Should not fall, even if disturbed

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Challenging case Bipedal Machines

• Should not fall, even if disturbed

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Challenging case Bipedal Machines

• Should not fall, even on shaky ground

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Challenging case Bipedal Machines

• Should not fall, even on shaky ground

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Challenging case Bipedal Machines

• Should not fall, even on shaky ground

• But there are limits!

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Approach walking task spec
Qualitative State Plan
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Computing torques to achieve a particular state
goal is challenging
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Hybrid executive and multivariable controller
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Hybrid executive coordinates controllers to
sequence plant through poses in qualitative state
plan
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Hybrid executive coordinates controllers to
sequence plant through poses in qualitative state
plan
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Hybrid executive coordinates controllers to
sequence plant through poses in qualitative state
plan
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Hybrid executive coordinates controllers to
sequence plant through poses in qualitative state
plan
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Hybrid executive coordinates controllers to
sequence plant through poses in qualitative state
plan
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• Multivariable controller
• makes state plan quantities, like CM, directly
  controllable
• allows hybrid executive to control CM by
  adjusting linear gain parameters

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Innovations

• Requirement Stable walking

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Innovations

• Requirement Stable walking

Previous Approaches
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Innovations

• Requirement Stable walking

Previous Approaches
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Innovations

• Requirement Stable walking
• How to get to the right place at the right time?
• What if terrain requires irregular foot
  placement?

Previous Approaches
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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
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Innovations

• Requirement ability to execute task-level plans
• How should walking plans be expressed?
• What are the requirements for successful plan
  execution?

Previous Approaches
Detailed actuated trajectory spec.
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Innovations

• Requirement ability to execute task-level plans
• How should walking plans be expressed?
• What are the requirements for successful plan
  execution?

Previous Approaches
Innovation
Detailed actuated trajectory spec.
Qualitative state trajectory spec.
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Innovations

• Requirement ability to execute task-level plans
• How should walking plans be expressed?
• What are the requirements for successful plan
  execution?

Previous Approaches
Innovation
Detailed actuated trajectory spec.
Qualitative control plan
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Innovations

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

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Innovations

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

Previous Approaches
Uses primarily ankle torque strategy
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Innovations

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

Previous Approaches
Innovation
Use three balance strategies
Uses primarily ankle torque strategy
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Humans use Three Balance Strategies

• Stance ankle torque

• Stepping

• Movement of non-contact segments

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Innovations

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

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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
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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
Innovation
Underactuated system - no equilibrium point (no
ability to wait)
Morris, 2001
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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.
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• Multivariable controller makes CM directly
  controllable

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Multivariable Controller Requirements

• Want to specify coarse setpoint
• Forward CM setpoint 0
• Lateral CM setpoint 0
• Controller should figure out detailed joint
  trajectories

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

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Hybrid Executive Requirements

• Multivariable controller accepts single setpoint

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Hybrid Executive Requirements

• Multivariable controller accepts single setpoint

• Cant, by itself, sequence through multiple
  setpoints

• Need hybrid executive for that

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At start of control epoch, hybrid exec. sets
controller gains
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Hybrid Executive guides each variable to its goal
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Hybrid Executive transitions to next epoch

• when goal for each variable is achieved

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What if there is a disturbance?

• trip recovery

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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

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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

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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?

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Plan compiler computes limits
Computes spatial and temporal regions for all
activities
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Plan compiler synthesizes controllers
Control info expressed as ranges on SISO
parameters
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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

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How does the plan compiler compute region limits,
synthesize controllers?

• Initial and goal regions

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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?

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How does the plan compiler compute region limits,
synthesize controllers?

• Lb fastest trajectory from slowest start
• Worst-case (slowest) start is point B

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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

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How does the plan compiler compute region limits,
synthesize controllers?

• Consider single acceleration spike as control
  input
• Spike occurs at beginning

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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)

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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

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How does the plan compiler compute region limits,
synthesize controllers?

• Spike of right size at end results in GST
  (Guaranteed Slowest Trajectory)

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Existence of controllable temporal range in goal

• If t(GFT)ltt(GST) then presence of trajectory in
  goal pos./vel. region can be guaranteed for any
  time t(GFT), t(GST)
• By adjusting spike

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GFT, GST with linear control law

• Adjust control law parameters to get GFT, GST

C max pos., min vel. (slowest finish) D min
pos., max vel. (fastest finish)
A max pos., vel. (fastest start) B min pos.,
vel. (slowest start)
Assume monotonic velocity

• Maximize controllable temporal range, initial
  region size
• Subject to limits on control inputs

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Controllable Regions for CM
1
2
Lateral
3
4
Forward
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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

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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

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Addendum
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Trade-off Between Region Size and Temporal Range
t(GFT) t(GST)
GFT in red, GST in blue, Nom in green
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Trade-off Between Region Size and Temporal Range
t(GFT) t(GST)
t(GFT) gt t(GST)
Some uncertainty in duration
GFT in red, GST in blue, Nom in green
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Strong and Dynamic Hybrid Controllability
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Strong and Dynamic Hybrid Controllability