Kinetostatic Design of an Articulated Leg-wheel Locomotion Subsystem

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Kinetostatic Design of an Articulated Leg-wheel
Locomotion Subsystem
Jun, Seung kook Advisor Dr. Venkat Krovi
Mechanical and Aerospace Engineering State
University of New York at Buffalo
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AGENDA

• Introduction
• Design Approach
• Kinetostatic Design of Single DOF Coupled Serial
  Chain(SDCSC)
• Kinetostatic Design of Four-bar Linkage
• Equilibrium Analysis
• Conclusion Future work

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Legged and Wheeled Systems
Introduction
Legged System
Climb step Cross ditches Walk on extremely rough
terrain Decoupled motion of the foot from the
motion of chassis
Leg-wheel System
Wheeled System
Dominated on prepared surfaces Provides passive
support Vehicle wheels simplifies the
actuation/control requirement
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Articulated Leg-wheel System (combining benefits)
Introduction
Shrimp, Bluebotics
Mars Rover, JPL
WorkPartner, Halme
RollerWalker, Hirose
Gofor,JPL
ALDURO, UDE
NOMAD, CMU
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Major Design Criteria
Introduction
Conflicting ?
Workspace
Achieving adequate ground clearance Ability to
surmount obstacle Open-loop linkages are
preferred
Suspension
Need an extremely small workspace Highly stiff
articulation Closed-loop linkages are preferred
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Specific Design Criteria In-chain Articulated
D.O.F
2-DOF
0-DOF
1-DOF
2-DOF
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Specific Design Criteria -Equilibration
Passively equilibrated configuration
Actively equilibrated configuration
Active Equilibration Torque Requirement
Equilibration for a 1-link mechanism
By changing configuration of legged system with
low power, one can save a lot of energy
consumption for supporting load
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Single D.O.F Articulated Leg-wheel System
Introduction
Proposed articulated leg-wheel system design
Fourbar-based configuration
Single Degree of Freedom Coupled Serial Chain
(SDCSC) based configuration (Krovi,1998)
Increasing Number of Articulation
Increasing Overall Control Complexity
Simple Lower-pair Joint
Limited Geometric Motion Capability
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Research Goals
Introduction
Creation, analysis and realization of the full
potential of these two candidate single D.O.F
designs for the articulated leg-wheel system.
Design an articulated leg-wheel system to match
the desired kinematic trajectories and static
torque profiles by use of methods from
kinetostatic synthesis and optimization
Examining static equilibrium depends critically
upon various mechanism parameters
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Applied Load at the End Effector
Design Approach
Schematic
FBD
Slant is 0 deg
X Reaction Torque by Motor Y Systems Weight
Slant is 90 deg

X Zero Y Motor Torque and Weight
End-effector forces with different motor torque
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Desired Kinematic Motion Curve
Design Approach
The principal reason for articulations to enable
the robot to accommodate and overcome most
obstacles by permitting the front wheel to rise
up when it collides with the obstacle.
Simple rising motion
Rising and offset motion
A designer has freedom in selecting the type,
number and locations of the precision points.
Open loop Desired Curve
Closed loop Desired Curve
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Desired Static Torque Design
Design Approach
Precision Torque
Desired Torque Curve
Stable/Unstable Equilibrium, Maximum/Minimum
Torque
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Kinetostatic Design Approach (Krovi,1998)
Design Approach
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Single Degree-of-freedom Coupled Serial Chain
Design/Synthesis of SDCSC
Two links SDCSC- based leg-wheel design
Three links SDCSC- based leg-wheel design
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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2-Link SDCSC Kinematic Synthesis
Design/Synthesis of SDCSC
Precision Point Constraints can be expressed as
By selecting
as the four free choices,
A linear solution of the equation can be obtained
as
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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3-Link SDCSC Kinematic Synthesis
Design/Synthesis of SDCSC
Precision Point Constraints can be expressed as
as the six free choices,
By selecting
A linear solution of the equation can be obtained
as
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Desired Trajectory
Design/Synthesis of SDCSC
Desired trajectory of end effector for 2-link
SDCSC
Desired trajectory of end effector for 3-link
SDCSC
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Kinematic Optimization
Design/Synthesis of SDCSC
Objective Function of 2-link SDCSC
Objective Function of 3-link SDCSC
subject to
subject to
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Kinematic Synthesis/Optimization Results
Design/Synthesis of SDCSC
Three link SDCSC configuration I
Three link SDCSC configuration II
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Kinematic Synthesis/Optimization Results
Design/Synthesis of SDCSC
Two link SDCSC configuration I
Three link SDCSC configuration I
Three link SDCSC configuration II
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Static Synthesis of SDCSC
Design/Synthesis of SDCSC
The angular extension of each spring is
Torsion springs at joints
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Virtual Work
Design/Synthesis of SDCSC
the actual torque requires to equilibrate the
system is computed as
The spring torque equation can also be rewritten
in terms of the relative angular motion of the
first joint as a linear equation as
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Precision Point Synthesis and Optimization
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Design/Synthesis of SDCSC
we only have two design variables for use in the
static synthesis process regardless of number of
links of SDCSC mechanism. These can be used in
the following ways.
Two static precision points and no free choices
For a two static precision position problem, the
spring constants are calculated by substituting
precision torques the specified torques
at the precision relative angle
This gives the designer considerable freedom for
selection of a spring constant by varying the
initial unloaded configuration
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Precision Point Synthesis and Optimization
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Design/Synthesis of SDCSC
For three link SDCSC case, the relationship
between the individual torsional spring
parameters k1, k2, k3 and the generalized lumped
spring parameters A,B can be expressed as
Solve for the unknown spring constants in the
least-square sense using pseudo-inverse of S
matrix.
No static precision points and using both
variables as free-choices
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Precision Point Synthesis and Optimization
Result
Design/Synthesis of SDCSC
Two static precision points and no free choices
No static precision points and using both
variables as free-choices
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Precision Point Synthesis and Optimization
Result
Design/Synthesis of SDCSC
Three link SDCSC configuration I
Precision Torque Synthesis
Desired Torque Curve Optimization
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Kinematic Synthesis
Design/Synthesis of Fourbar
From loop-closure equation,
Two Precision Points
Three Precision Points
Where, M and N are position vector of first link
fixed on the chassis.
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Kinematic Optimization
Design/Synthesis of Fourbar
Objective Function ( 2 Precision Points)
subject to
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Kinematic Synthesis Result
Design/Synthesis of Fourbar
Two Precision Points Open Desired Curve
Two Precision Points Closed Desired Curve
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Synthesis of Fourbar
Design/Synthesis of Fourbar
The sum of virtual work done by the external
forces, reaction torque at joint and spring
forces is equal to zero. Thus reaction torque at
driving joint is
FBD of Fourbar
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Synthesis of Fourbar 1
Design/Synthesis of Fourbar
We have four design variables for use in the
static synthesis process of fourbar mechanism.
These can be used in the following ways.
Four static precision points and no free choices
For a four static precision position problem, the
spring constants are calculated by substituting
precision torques the specified torques
at the corresponding driving angle
This gives the designer considerable freedom for
selection of a spring constant by varying the
initial unloaded configuration
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Synthesis of Fourbar 2
Design/Synthesis of Fourbar
No static precision points and using four
variables as free-choices
Four static precision points and no free choices
No static precision points and using four
variables as free-choices
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Equilibrium Path
Static Equilibrium Analysis
If the path represents configurations of static
equilibrium it is called an equilibrium path (and
each point is called an equilibrium point).
Stable Equilibrium If the system always returns
to it after small disturbance Unstable
Equilibrium If the system moves away from the
equilibrium after small disturbance
The sign of the tangent stiffness is closely
associated with the question of stability of an
equilibrium state
A positive tangent is necessarily associated with
unstable equilibrium.
A negative tangent is necessary but not
sufficient for stability.
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Static Equilibrium Analysis
Method for Equilibration
Passive balancing by adding counterweights or
using springs.
Counterweight
Passive counterbalancing has been done by adding
masses, off-setting a system mass with another
mass creating an opposite moment.
Spring-equilibrium System
Using the stored energy in springs to counter the
effects of gravity
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Equilibrium Analysis for Spring
Static Equilibrium Analysis
The sum of the forces must equal to zero, and the
sum of the torques must equal zero
Link will rotate link counter-clockwise direction
Link will rotate link clockwise direction
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Equilibrium Analysis for Spring
Static Equilibrium Analysis
Modification of Location of Equilibrium One
Link Example
Changing Preload Angle
Changing Spring Constant
Changing Motor Torque
by changing constants, we can modify the
locations and type of equilibrium.
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Equilibrium Analysis for Spring
Static Equilibrium Analysis
Basin of Attraction Optimization One Link
Example
One of examples of objective function can be
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Equilibrium Analysis for Counterweight
Static Equilibrium Analysis
By changing vector from the pivot we can shift
curve
By changing weight we can change amplitude of
curve
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Equilibrium Analysis for SDCSC
Static Equilibrium Analysis
Precision Torque Design
Equilibrium Position
Kinematic Configuration
Two Precision Torques
Enhance Stiffness, Equilibrium Design
First Precision Torque is zero at joint angle 5
deg
Second Precision Torque is positive at joint
angle 32 deg
Enhance reconfigurability
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Equilibrium Analysis for Fourbar
Static Equilibrium Analysis
Equilibrium Position
Kinematic Configuration
Two Precision Torques
First Precision Torque is negative at joint angle
0 deg
Enhance Stiffness,
Second Precision Torque is zero at joint angle 28
deg
Equilibrium Design
Enhance reconfigurability
Third Precision Torque is positivee at joint
angle 125 deg
Fourth Precision Torque is zero at joint angle 32
deg
Equilibrium Design
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Simulation of Two link SDCSC with Solidworks and
VisualNastran
Simulation
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Research Contribution
Conclusion

• Systematic evaluation of the nature of the
  articulations and affiliated hardware constraints
  in achieving the desired workspace and the
  desired suspension characteristics.
• Adaptation of the kinematic and kinetostatic
  design procedure developed for dimensional
  synthesis of mechanisms for designing articulated
  leg-wheel system.
• Application of this modified dimensional
  synthesis process to assist in the selection of
  the optimal design of four bar and SDCSC-based
  articulated leg-wheel systems to guide the wheel
  axle through several positions while supporting
  external loads.
• Careful examination of the role of structural-
  and spring-assisted equilibration for
  minimization of the actuation requirements to
  support the external loads for both four-bar and
  SDCSC based designs.
• Analysis of the role of various mechanism
  parameters on locations, stability and basins of
  attraction of the static system equilibria in the
  articulated leg-wheel designs.

Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion
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Future Work
Conclusion
Virtual and physical prototype test Application
of nonlinear spring Considering link weight and
friction Spatial motion analysis Dynamic behavior
of the device Enhancing configurability by
active/semi-active device
Introduction Design Approach SDCSC Fourbar
Equilibrium Analysis Conclusion