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«by GABRIEL MARTIN NELSON Submitted in partial fulfillment of the requirements For the degree of Doctor of Philosophy Thesis Advisor: Dr. Roger D. ...»

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LEARNING ABOUT CONTROL OF LEGGED LOCOMOTION USING A

HEXAPOD ROBOT WITH COMPLIANT PNEUMATIC ACTUATORS

by

GABRIEL MARTIN NELSON

Submitted in partial fulfillment of the requirements

For the degree of Doctor of Philosophy

Thesis Advisor: Dr. Roger D. Quinn

Department of Mechanical and Aerospace Engineering

CASE WESTERN RESERVE UNIVERSITY

May, 2002 Copyright © 2002 by Gabriel Martin Nelson All rights reserved

CASE WESTERN RESERVE UNIVERSITY

SCHOOL OF GRADUATE STUDIES

We hereby approve the dissertation of

GABRIEL MARTIN NELSON

candidate for the Doctor of Philosophy degree *.

Committee Chair: ________________________________________________

Dr. Roger D. Quinn Dissertation Advisor Professor, Department of Mechanical and Aerospace Engineering Committee: _____________________________________________________

Dr. Joseph M. Mansour Professor, Department of Mechanical and Aerospace Engineering Committee: _____________________________________________________

Dr. Stephen M. Phillips Professor, Department of Electrical Engineering and Computer Science Committee: _____________________________________________________

Dr. Roy E. Ritzmann Professor, Department of Biology May, 2002 *We also certify that written approval has been obtained for any proprietary material contained therein.

Waiver of Reproduction Rights To Stephanie Do you see a man wise in his own eyes?

There is more hope for a fool than for him.

Proverbs 26:12 For the LORD gives wisdom, and from His mouth come knowledge and understanding.

Proverbs 2:6

Table of Contents Table of Contents

List of Tables

List of Figures

Acknowledgements

Abstract

1 Introduction

1.1 Robot 3

1.2 Thesis Outline

Works Cited

2 Review

Works Cited

3 Posture Control

3.1 Single Leg Mechanics

3.2 Somatosensory Feedback of Body Position

3.3 Multi-leg Mechanics

3.4 Solving the Force Distribution Problem

3.5 Results

3.6 Conclusions

Works Cited

4 Posture Control Implementation Issues

4.1 Original Hardware Setup

4.2 Open-loop force control

4.3 Conclusion

Works Cited

5 Inverse kinematics

5.1 General Notation

5.2 Redundant limb kinematics

5.3 A Practical Goal: Maximize Leg Mobility

5.4 A Simple Redundant Manipulator

5.5 Ways of finding equilibrium solutions for both the SRM and Robot 3....63

5.6 A Biological Perspective: Animal-Like Movements

5.7 Neural-Network Implementation

5.8 Results and Conclusions

Works Cited

6 Local Control Implementation Details

6.1 Improvements in the Robot 3 control system

–  –  –

7 Conclusions and Future Work

7.1 Future Work

7.1.1 Open-loop control issues

7.1.2 What about those strain gages?

7.1.3 Lead compensation

7.1.4 Positive load feedback

7.1.5 Revalving the robot

7.2 Conclusions

7.3 Some lessons learned from Robot 3

7.3.1 Carefully consider using pneumatic pulse actuation

7.3.2 Carefully consider which sensors to use

7.3.3 Simulate

7.3.4 Strive for modularity and ruggedness

7.3.5 Successful gait coordination is not the same as successful walking130 7.3.6 Developing the basic control system takes time

7.3.7 Acoustic aesthetics matter

Works Cited

Appendix

Bibliography

–  –  –

Figure 2: Strain gages on a Robot 3 leg. Three pairs of strain gages measure bending at three different points on the robot leg. The In-Plane-Femur (IPF) gages at the proximal end of the femur, and the In-Plane-Tibia (IPT) gages at the proximal end of the tibia, measure bending about axes perpendicular to the plane formed by the femur and tibia. The Out-of-Plane-Tibia (OPT) gages measure bending at the proximal end of the tibia about an axis lying in the plane formed by the femur and tibia.

Figure 3: Whegs. A reduced-actuation hexapod vehicle built at the CWRU BioRobotics Lab. A single motor drives six “whegs” – or wheel-legs – which are phased in a nominal tripod gait. A torsional compliance in the drive shaft at each wheg allows for phasing variations such as those needed for climbing obstacles. Whegs moves at about three bodylengths per second (60 in/s), and can climb barriers greater than its body height

Figure 4: Protobot is an 18 DOF pneumatically actuated hexapod robot modeled after Periplaneta americana.

Figure 5: RHex is a typical example of a Type 3 robot. Each leg is a single compliant spoke, driven, at the hip, by a motor under PD control. The spoke swings in a full circle parallel to the sagittal plane of the robot.

RHex is a simple and reliable hexapod robot with good rough terrain capabilities





Figure 6: Single leg mechanics notation. The body reference frame consists of the x 1 ≡ rostral or direction of travel, y 1 ≡ lateral or left, z1 ≡ vertical.

Each body (1 through 4) contains its own body-fixed reference frame.

The vector definitions (Li and W i) are given in the text

Figure 7: Single virtual leg model of robot mechanics. Provided with desired virtual forces acting on the body, the posture controller predicts the position of a center of pressure (COP) where a virtual leg would, at that instant, produce those virtual forces. The scheme is repeated for the xz plane. The virtual leg would also have a foot (not shown) that would produce a desired virtual M z.

ix Figure 8: Disturbance rejection while standing via posture control. While standing, the robot was shoved repeatedly. Each arrow indicates a disturbance. The robot swayed and returned to a nominal standing position. “y pos” indicates the y, or lateral, position (see Figure 6) of the body. “y cop” is the y location of the COP. The COP moved to counteract the disturbances. The COP was slightly negative because the robot perceived a small roll error (lean to left). Stiction in the cylinders caused the initial and final body positions to be slightly different.

Figure 9: Vertical load transfer to move COP. Corresponding to Figure 8, this figure shows how vertical load responsibility was transferred to the left side legs as the COP moves to counteract the disturbances. “left/right n” is the ipsilateral sum of n l values

Figure 10: Robot 3 lifts a 30 pound payload. The payload, which is equivalent to its own weight, is suspended below the robot with cables. The robot is able to perform “push-ups” while doing this.

Figure 11: The overall original posture control system. Computer #2 (PC#2), which was slaved to computer #1 (PC#1), performed PWM on the 48 valves at a frequency of 50 Hz according to commanded duty cycles from PC#1. PC#1, directly reading sensory information from the physical robot, performed posture control calculations and output commanded duty cycles to PC#2

Figure 12: Original Robot 3 posture control basic electronics. PC#1 was a 133MHz Pentium desktop running the posture controller in DOS. The controller dispatched (unbuffered) commanded duty cycles to PC#2, a 127MHz AMD desktop, via 115200 bps serial communication. PC#1 also read 24 potentiometer signals from an ISA A/D (12 bit) card (original gage signals, which were not used, were one per leg and meant for binary contact sensing only). PC#2, running in DOS, polled a timer card to perform 50 Hz PWM with about a 1% resolution. A digital I/O card drove 48 opto-isolators, which in turn drove the valves of the robot.

Figure 13: Original duty cycle to force output relationship at 50 Hz PWM.

Various cylinder sizes were tested for their steady-state force output as a function of duty cycle at 50 Hz PWM. The curves were then normalized by the theoretical maximum force for that cylinder size, and a sigmoidal curve was fit to the data (Eq. (20)). The inverse of this relationship was used for open-loop force control

Figure 14: An extremely simple redundant manipulator (SRM). Joint-space DOF are θ1 and θ2 while there is a single task-space DOF, x d. θ1 is measured from the horizontal and θ2 is relative to link 1 (positive rotations are

–  –  –

Figure 15: Solution manifolds to the forward kinematics of the SRM with l1 = l2 = 1. The xd = 2 manifold (which is just a point) is located at {θ1, θ2} = {0, 0}. The xd = -2 (also just a point) is located at {π, 0}. The remaining manifolds are evenly spaced in task-space at 0.2 intervals between –2 ≤ xd ≤ 2, such that traversing a straight line from {0, 0} to {π, 0}, we cross the following manifolds: x d = 1.8, 1.6, … 0, … -1.6, The entire pattern is repeated at 2π intervals in each direction (making the point {-π, 0} also a x d = -2 manifold)

Figure 16: Loci of all possible solutions to Eq. (33) for the SRM with k1 = k2 = 1, θ1 = π/2, θ2 = -π/2. The reachable joint-space has been limited to 0 θ1 π and –π θ2 0 in order to exclude any structural singularities.

The two loci have been distinguished as “primary” and “secondary”.

The arrow indicates the approximate location on the secondary locus where the augmented Jacobian (see text) becomes singular. Refer to Figure 15 for the x d manifold values.

Figure 17: Comparison of three inverse kinematic solution methods applied to the SRM. All parameters are set as discussed in Figure 16, C j is the identity matrix, and the starting position is the unloaded equilibrium position. The goal is to move the end effector from the unloaded equilibrium position (x d = 1) to x d = -1. The quasi-static method is based on a straightforward scheme which simulates the motion of the manipulator quasi-statically with joint-space springs and dampers while an applied force at the end effector drives the system to desired taskspace positions. The Seraji method is based on an online redundant manipulator control scheme that controls end effector motion in taskspace as well as one or more user-defined “self-motion” task functions.

The minimum joint-space velocity method is the standard MoorePenrose pseudoinverse of the Jacobian with a constant weight matrix.

Exact methods are not shown, since they lie directly along the primary locus

Figure 18: Block diagram of Seraji redundant manipulator control. An augmented task-space vector, Y, consisting of end effector position as well as (N-M) task functions, is fed back for PD control to produce a corrective augmented wrench vector, FA. Augmented Jacobians transform FA into joint torques, τ, and joint positions and velocities into xi augmented task-space velocity. KP and KD are NxN matrices of proportional and derivative gains respectively.

Figure 19: Partial overhead view of Robot 3 showing left front leg x-y coordinate system used for inverse kinematics studies. The origin is the bodycoxa joint, which is where the leg attaches to the body. This view looks in the –z direction.

Figure 20: Desired circular foot path vs. neural-network output. The trained neural-network outputs joint angles for desired foot positions along a 5 inch radius circular path in the x-y plane, with p foot/d,z = -6 inches below the body-coxa joint. These joint angles result in a slightly distorted actual foot position path.

Figure 21: Desired square foot path vs. neural-network output. The trained neural-network outputs joint angles for desired foot positions along a 7x7 inch square path in the x-y plane, with p foot/d,z = -6 inches below the body-coxa joint. These joint angles result in a slightly distorted actual foot position path.

Figure 22: Neural-network to cockroach comparison. Forceps were used to move the tibia-tarsis joint of a deinnervated left front cockroach leg through a walking cycle motion. The motion was filmed and digitized to produce joint angle trajectories (dark lines) and scaled foot motion. This foot motion was then used as input to the neural-network to produce optimal joint angle trajectories for comparison (light lines). The poorer fit of the FT trace may have two possible sources: the joint compliance functions were too simple, or the forceps applied an unknown environmental moment to the tibia.

Figure 23: Snapshots of Robot 3 air-walking in a tripod gait. The sequence of snapshots, cropped from digital video of the robot, goes from left to right, top to bottom. In the first snapshot, the near-side middle leg is in stance while the front and rear legs transition to swing. In the second snapshot, the front and rear legs are in full swing. In the third snapshot, they transition into stance, while in the fourth snapshot, the middle leg transitions to swing.

Figure 24: Improved basic control system setup. PC#1 (500MHz Pentium), which runs the high level controller (HLC), performs longer latency, higher complexity calculations (such as posture control) and issues any of a variety of commands or guidelines to PC#2 (127MHz AMD) which is running the low level controller (LLC). The HLC is also responsible for setting LLC parameters such as control loop gains and set-points. The LLC performs all data acquisition of sensor feedback and commands the valves of the robot. It also directly copies sensor data to the HLC

xii Figure 25: Control system communications structure. The HLC consists of two processes: a lower priority background process running the actual HLC code and a higher priority communications ISR (Com ISR). The LLC consists of three processes: a lower priority background process that performs certain non-time-critical calculations as well as moving data to and from the Com ISR buffers and memory, a middle priority process (the Com ISR), and a high level PWM ISR which does elementary control calculations, A/D conversions and valve commanding. One problem with this setup is that the flow of data to and from the PWM ISR to the HLC is handled by the low priority background process.



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