Sprawl Robots Biomimetic Design Analysis: Simplified Models Motion Analysis Performance Testing

Design Inspiration Control heirarchy Passive component Active component

Is Passive Enough? Passive Dynamic Stabilization No active stabilization Geometry Mechanical system properties

Sprawl 1.0: Biomimetic, not just a copy Low-Level Control MURI Sprawl 1.0: Biomimetic, not just a copy - cockroaches don’t attempt to use complicated, high-level control but still outperform robots that do (Dante - very impressive, but took a very long time to cross rough terrain because of footholds), not searching for footholds, no real-time solutions to optimal force vectors, what else? - sprawled posture and decoupled leg mechanism biologically selected for locomotion - sprawled posture has a lower center of mass which is harder to tip - decoupled mechanism allows for a simplified control - thrust force is the direct result of one actuator instead of a combination of actuators (main power producing muscles have been identified) - mechanical system rather than high-level control rejects disturbances Full’s research highlights certain important locomoting components Power-producing thrust muscles Supporting/repositioning hip joints

Implementation Cockroach Geometry Functional Biomimesis Shape Deposition Manufactured Robot Passive Compliant Hip Joint Effective Thrusting Force Rotary Joint Prismatic Joint Damped, Compliant Hip Flexure Embedded Air Piston

Sprawlita Mass - .27 kg Dimensions - 16x10x9 cm Leg length - 4.5 cm Max. Speed - 39cm/s 2.5 body/sec Hip height obstacle traversal

Mechanical System Properties Prototype: Empirically tuned properties Design for behavior Understanding ? Mechanical System Properties

Robot Analysis for Design Simplified Models Motion Analysis Performance Testing 24 deg. k, b, nom

Robot Analysis for Design Simplified Models Motion Analysis Performance Testing 24 deg. k, b, nom

“Simple” Model 18 parameters to tune - TOO MANY! K, B, nom k, b, nom Full 3D model Planar model Symmetry assumption k, b, nom Body has 3 planar degrees of freedom x, z, theta mass, inertia 3 massless legs (per tripod) rotating hip joint - damped torsional spring prismatic leg joint - damped linear spring 6 parameters per leg 18 parameters to tune - TOO MANY!

Simplest Locomotion Model g g k, b, nom Biped Biped Quadruped Body has 2 planar degrees of freedom x, z mass 4 massless legs freely rotating hip joint prismatic leg joint - damped linear spring 3 parameters per leg 6 parameters to tune, assuming symmetry

Modeling assumptions g Time-Based Mode Transitions One “reset” mode Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression Leg Set 2 1 State Time x = state trajectory Stride Period T 1 McMahon, et al 1987

Modeling assumptions t = 2T- g Time-Based Mode Transitions Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression t = 2T- State x Leg Set Leg Set Leg Set Leg Set 1 2 1 2 T Time = state trajectory Stride Period 1 McMahon, et al 1987

Modeling assumptions t = 2T+ g Time-Based Mode Transitions Leg Set 2 1 State Time x = state trajectory Stride Period t = 2T+ T g Time-Based Mode Transitions Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression 1 McMahon, et al 1987

Modeling assumptions t = 2T + 1/3T g Time-Based Mode Transitions Leg Set 2 1 State Time x = state trajectory Stride Period t = 2T + 1/3T T g Time-Based Mode Transitions Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression 1 McMahon, et al 1987

Modeling assumptions t = 2T + 2/3T g Time-Based Mode Transitions Leg Set 2 1 State Time x = state trajectory Stride Period t = 2T + 2/3T T g Time-Based Mode Transitions Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression 1 McMahon, et al 1987

Modeling assumptions t = 3T- g Time-Based Mode Transitions Leg Set 2 1 State Time x = state trajectory Stride Period t = 3T- T g Time-Based Mode Transitions Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression 1 McMahon, et al 1987

Modeling assumptions t = 3T+ g Time-Based Mode Transitions Leg Set 2 1 State Time x = state trajectory Stride Period t = 3T+ T g Time-Based Mode Transitions Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression 1 McMahon, et al 1987

Modeling assumptions t = 3T + 1/3T g Time-Based Mode Transitions Leg Set 2 1 State Time x = state trajectory Stride Period t = 3T + 1/3T T g Time-Based Mode Transitions Clock-driven motor pattern “Groucho running”1 One “reset” mode Two sets of legs - Two modes Symmetric - treat as one mode Mode initial conditions Nominal leg angles Instant passive component compression 1 McMahon, et al 1987

Non-linear analysis tools Discrete non-linear system Fixed points numerically integrate to find exclude horizontal position information = state trajectory = fixed points xk+1 = xk = x* Leg Set 2 1 State Time x = state trajectory Stride Period T

Non-linear analysis tools Floquet technique Analyze perturbation response Digital eigenvalues via linearization - examine stability Use selective perturbations to construct M matrix = nominal trajectory Numerically Integrate

Analysis trends Relationships Use for design 6.5 7 7.5 8 8.5 9 9.5 10 1.2 1.4 1.6 1.8 2 2.2 2.4 2.6 2.8 0.04 0.045 0.05 0.055 0.06 0.065 0.07 0.075 Damping (N-s/m) Recovery Rate Horizontal Velocity X_dot (m/s) 1/max[eig(M)] Relationships damping vs. speed and “robustness” stiffness, leg angles, leg lengths, stride period, etc Use for design select mechanical properties select other parameters Insight into the mechanism of locomotion

Locomotion Insight Body tends towards equilibrium point Parameters and mechanical properties determine how

Design Procedure Find parameter set that will yield fixed points Establish trends by varying one parameter Perturb and integrate Build the M matrix Find eigenvalues and performance index Select new parameter value Iterate

Design Example Robustness Speed Damping Damping Damping Stiffness Notice that with the damper there is a tradeoff btw speed and robustness, but hat tradeoff does not exist by changing the springs In other words, you sacrifice speed for robustness and vice versa in the trends with the dampers, while this is not true when changing the spring rates. Stiffness Stiffness Stiffness Speed = 0 Speed = 13 cm/s Speed = 23.5 cm/s

Results and Future Work “Biomimetic” locomotion Feedforward motor program Preflexes Geometry Good biomimetic locomotion is more subtle Speed without sacrificing robustness Robustness without sacrificing efficiency Adaptation useful Changes in “global” conditions Fast and Robust

Good biomimetic locomotion Comparing detailed results to cockroach locomotion data Ground reaction forces Leg workloops Efficiency 3 legged model Different than 2 legs – more freedom! Faster & More Efficient

Good biomimetic locomotion Comparing detailed results to cockroach locomotion data Ground reaction forces Leg workloops Efficiency 3 legged model Different than 2 legs – more freedom! Time (seconds) Vertical Displacement Vertical Displacement Time (seconds) Middle leg = 72.5 degrees Middle leg = 70 degrees

Need for Adaptation Robustness, speed, and efficiency are sensitive Model parameters Geometry (leg angles, lengths) Relative stiffnesses Number of legs Environment Slope Slope Leg angle Velocity

Robot Analysis for Design Simplified Models Motion Analysis Performance Testing 24 deg. k, b, nom

Motion Analysis Compare simple models to cockroach kinematic data Horizontal plane model (O)

Motion Analysis Experiments in finding model parameters to match kinematic data (O)

Motion Analysis Extract passive stabilizing properties Horizontal plane model

Motion Analysis Different set of model parameters will result in different performance (O) (O)

Motion Analysis Hi-speed motion capture of robot to qualify performance (O)

Motion Analysis Results show effect of system parameters in resulting motion Center of Mass Sagittal Trajectories -0.08 4.35 Hz -0.082 -0.084 -0.086 vertical (m) -0.088 7.7 Hz -0.09 -0.092 -0.094 12.5 Hz -0.096 -0.098 0.05 0.1 0.15 0.2 0.25 horizontal position (m)

Motion Analysis Integrate motion data with on-board instrumentation

Robot Analysis for Design Simplified Models Motion Analysis Performance Testing 24 deg. k, b, nom

Work to Date Measurements of: Maximum Velocity Maximum Obstacle Clearance

Reasons for Testing Measure performance of system while varying: System parameters Terrain properties Understand locomotion Adapt to environment Long-term durability

Testing Conditions Vary Terrain Properties Slope Roughness Properties Smooth Fractal Properties Packed Dirt Gravel Sand

Velocity vs. Slope

Multivariable Testing Many Parameters affect Performance Duty Cycle Gait Period Front Leg Angles Middle Leg Angles Back Leg Angles Center of Mass Pressure Mass Compliance Leg Length Body Length Etc.

Velocity vs. Slope and Gait Period

Velocity vs. Slope and Gait Period

Velocity vs. Slope and Gait Period

Velocity vs. Slope and Duty Cycle

Velocity vs. Slope and Duty Cycle

Leg Testing Performance is heavily dependent on the combination of leg angles Therefore, the leg angles cannot be independently examined. Begun Factorial Testing Test relationships between various parameters Leg Angles, Duty Cycle, Gait Period Compliance Define Factorial Testing while talking Mention that FT is not yet complete. It “will allow us to” …

Future Work Test other parameters for maximum velocities Analyze data for effect Incorporate findings Table of best conditions Adaptive code Progress to other terrains

Lessons Thus Far Ideal parameters change with slope Performance is dependent on the parameters and their interactions Adaptation increases capability