Symbolic Computation of the Lie Algebra se(3) of the Euclidean Group SE(3): An Application to the Infinitesimal Kinematics of Robot Manipulators
Abstract
:1. Introduction
2. Theory
2.1. Plücker Coordinates
2.2. Chasles’ Theorem
- The velocity component along the Mozzi axis is the same for all points on the body m.
- The velocity component of any point of the body m along an axis perpendicular to the ISA is null.
- The velocity vector of any point of the body m lies in a plane defined by two vectors; the first one is parallel to the ISA, while the second one is normal to the ISA and to the position vector of the point of interest and an arbitrary point on the ISA.
2.3. The Lie Algebra of the Euclidean Group
- Addition (+)The operation of addition (+) has the following properties:
- (a)
- Commutative ;
- (b)
- Associative ;
- (c)
- Identical additive , ;
- (d)
- scalar multiplication, . .
- Lie bracketThe Lie bracket has the following operations
- (a)
- Anti-symmetry (anticommutativity),
- (b)
- Alternativity (nilpotent), ;
- (c)
- Bilinearity, ;
- (d)
- Jacobi identity, ;
- (e)
- product rule, .
- Bilinearity
- (a)
- Klein form , ;
- (b)
- Killing form , .
2.4. Open Kinematic Chains
2.4.1. Velocity
2.4.2. Acceleration
3. Symbolic Computation: Maple Sheets
4. Examples
4.1. Planar RPR Serial Manipulator
4.2. PUMA Robot
4.2.1. Position Analysis
4.2.2. Velocity Analysis
4.2.3. Acceleration Analysis
4.2.4. Numerical Application
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
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Link Offset [mm] | Joint Angle | Link Length [mm] | Twist Angle [rad] |
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Gallardo-Alvarado, J.; Garcia-Murillo, M.A.; Tabares-Martinez, J.M.; Sandoval-Castro, X.Y. Symbolic Computation of the Lie Algebra se(3) of the Euclidean Group SE(3): An Application to the Infinitesimal Kinematics of Robot Manipulators. Mathematics 2024, 12, 2538. https://doi.org/10.3390/math12162538
Gallardo-Alvarado J, Garcia-Murillo MA, Tabares-Martinez JM, Sandoval-Castro XY. Symbolic Computation of the Lie Algebra se(3) of the Euclidean Group SE(3): An Application to the Infinitesimal Kinematics of Robot Manipulators. Mathematics. 2024; 12(16):2538. https://doi.org/10.3390/math12162538
Chicago/Turabian StyleGallardo-Alvarado, Jaime, Mario A. Garcia-Murillo, Juan Manuel Tabares-Martinez, and X. Yamile Sandoval-Castro. 2024. "Symbolic Computation of the Lie Algebra se(3) of the Euclidean Group SE(3): An Application to the Infinitesimal Kinematics of Robot Manipulators" Mathematics 12, no. 16: 2538. https://doi.org/10.3390/math12162538