Limiting-case Analysis of Continuum Trunk ... - Semantic Scholar
Limiting-case Analysis of Continuum Trunk Kinematics Bryan A. Jones and Ian D. Walker [email protected] [email protected] at
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Electrical & Computer Engineering
Tongues, trunks, and tentacles Muscular hydrostats: • Ability to grasp a wide variety of sizes and shapes • Compliant • Dexterous • Extensive sensory abilities • Able to explore under rocks, reach in holes
Smith, K.K. & W. M. Kier (1989) Amer. Sci. 77: 28-35.
Electrical & Computer Engineering
Biologically-inspired robotics Muscular hydrostats: • Ability to grasp a wide variety of sizes and shapes • Compliant • Dexterous • Extensive sensory abilities • Able to explore under rocks, reach in holes
Electrical & Computer Engineering
Applications • Remote exploration – Search and rescue / disaster relief
• Medical – Endoscope, steerable needles
• Space
R. R. Murphy, “Trial by Fire”
– Planetary surface exploration
• Military –
team
• Nuclear – Hazardous materials handling
Electrical & Computer Engineering
Modeling • Trunk physics produce constant curvature curves • Many continuum models view trunk as an arc – – – – – – –
Chen et al Sears and Dupont Webster et al Thomann et al Hannan and Walker Bailly and Amirat Jones and Walker Electrical & Computer Engineering
Limiting cases • No traditional singularities • The arc radius r for a straight trunk = ∞. – End effector position is ⎡ r (1 − cos sr −1 ) cos φ ⎤ ⎢ ⎥ ⎡0⎤ lim = ⎢ r (1 − cos sr −1 ) sin φ ⎥ = ⎢0 ⎥ ⎢ ⎥ r →∞ ⎢ ⎥ r sin sr −1 ⎢ ⎥ ⎢⎣ s ⎥⎦ ⎣ ⎦ – Numerically problematic (divide by 0, stability near limiting point)
Electrical & Computer Engineering
Solution approaches • Ignore / avoid – Lose workspace – Velocity difficulties – Uncertain area of instability
• Analyze – Analytical improvements – Define error metric – Determine stable region
Electrical & Computer Engineering
Analytical improvements • Kinematics from actuator length l1-3 complex. • Common denominator term is g = l12 + l2 2 + l32 − l1l2 − l2l3 − l1l3 (distance2 from l1=l2=l3) • Rewriting dramatically improves accuracy (10−7 vs. 10−16): g = ( l2 − l1 ) + ( l3 − l1 ) − ( l2 − l1 ) ( l3 − l1 ) 2
2
−308 ε = 2.26 ⋅ 10 • Also perturb with
Electrical & Computer Engineering
Kinematics results • Forward kinematics works for all l! • Velocity kinematics does not; g ≈ 0 → singularity. – Keep g > δ: when g = l12 + l2 2 + l32 − l1l2 − l2l3 − l1l3 < δ perturb l1 just enough to move outside region of instability by choosing l1 p = l2 + l3 ± 6l2l3 + 4δ − 3l2 2 − 3l32
• Find δ numerically
δ
Electrical & Computer Engineering
Numerical analysis •
Two steps
trunk straightness
•
||Jlimit – Jexact||
1. Develop an exact solution 2. Evaluate machineprecision at various δ
Exact solution: should approach limit – Compute Jexact using high-precision arithmetic (60 decimal digits) – Check against limit Electrical & Computer Engineering
Evaluate various δ ||Jexact – Jmachine||
Ideal δ D: approaching limit C: δ too large B: δ too small
δ k n u tr
ss e tn h ig a r st Electrical & Computer Engineering
Results and conclusions • Limiting-case numerical problems solved! • Rewrite g and add ε.
g = ( l2 − l1 ) + ( l3 − l1 ) − ( l2 − l1 ) ( l3 − l1 ) 2
2
• Compute A. • If g < δ, perturb l1: l1 p = l2 + l3 ± 6l2l3 + 4δ − 3l2 2 − 3l32
(find δ numerically) • Compute J.
Electrical & Computer Engineering
Thank you!
See www.ece.msstate.edu/~bjones or e-mail [email protected] for more information at
at
Electrical & Computer Engineering
at
at
at
Electrical & Computer Engineering
Tongues, trunks, and tentacles Muscular hydrostats: • Ability to grasp a wide variety of sizes and shapes • Compliant • Dexterous • Extensive sensory abilities • Able to explore under rocks, reach in holes
Smith, K.K. & W. M. Kier (1989) Amer. Sci. 77: 28-35.
Electrical & Computer Engineering
Biologically-inspired robotics Muscular hydrostats: • Ability to grasp a wide variety of sizes and shapes • Compliant • Dexterous • Extensive sensory abilities • Able to explore under rocks, reach in holes
Electrical & Computer Engineering
Applications • Remote exploration – Search and rescue / disaster relief
• Medical – Endoscope, steerable needles
• Space
R. R. Murphy, “Trial by Fire”
– Planetary surface exploration
• Military –
team
• Nuclear – Hazardous materials handling
Electrical & Computer Engineering
Modeling • Trunk physics produce constant curvature curves • Many continuum models view trunk as an arc – – – – – – –
Chen et al Sears and Dupont Webster et al Thomann et al Hannan and Walker Bailly and Amirat Jones and Walker Electrical & Computer Engineering
Limiting cases • No traditional singularities • The arc radius r for a straight trunk = ∞. – End effector position is ⎡ r (1 − cos sr −1 ) cos φ ⎤ ⎢ ⎥ ⎡0⎤ lim = ⎢ r (1 − cos sr −1 ) sin φ ⎥ = ⎢0 ⎥ ⎢ ⎥ r →∞ ⎢ ⎥ r sin sr −1 ⎢ ⎥ ⎢⎣ s ⎥⎦ ⎣ ⎦ – Numerically problematic (divide by 0, stability near limiting point)
Electrical & Computer Engineering
Solution approaches • Ignore / avoid – Lose workspace – Velocity difficulties – Uncertain area of instability
• Analyze – Analytical improvements – Define error metric – Determine stable region
Electrical & Computer Engineering
Analytical improvements • Kinematics from actuator length l1-3 complex. • Common denominator term is g = l12 + l2 2 + l32 − l1l2 − l2l3 − l1l3 (distance2 from l1=l2=l3) • Rewriting dramatically improves accuracy (10−7 vs. 10−16): g = ( l2 − l1 ) + ( l3 − l1 ) − ( l2 − l1 ) ( l3 − l1 ) 2
2
−308 ε = 2.26 ⋅ 10 • Also perturb with
Electrical & Computer Engineering
Kinematics results • Forward kinematics works for all l! • Velocity kinematics does not; g ≈ 0 → singularity. – Keep g > δ: when g = l12 + l2 2 + l32 − l1l2 − l2l3 − l1l3 < δ perturb l1 just enough to move outside region of instability by choosing l1 p = l2 + l3 ± 6l2l3 + 4δ − 3l2 2 − 3l32
• Find δ numerically
δ
Electrical & Computer Engineering
Numerical analysis •
Two steps
trunk straightness
•
||Jlimit – Jexact||
1. Develop an exact solution 2. Evaluate machineprecision at various δ
Exact solution: should approach limit – Compute Jexact using high-precision arithmetic (60 decimal digits) – Check against limit Electrical & Computer Engineering
Evaluate various δ ||Jexact – Jmachine||
Ideal δ D: approaching limit C: δ too large B: δ too small
δ k n u tr
ss e tn h ig a r st Electrical & Computer Engineering
Results and conclusions • Limiting-case numerical problems solved! • Rewrite g and add ε.
g = ( l2 − l1 ) + ( l3 − l1 ) − ( l2 − l1 ) ( l3 − l1 ) 2
2
• Compute A. • If g < δ, perturb l1: l1 p = l2 + l3 ± 6l2l3 + 4δ − 3l2 2 − 3l32
(find δ numerically) • Compute J.
Electrical & Computer Engineering
Thank you!
See www.ece.msstate.edu/~bjones or e-mail [email protected] for more information at
at
Electrical & Computer Engineering
