Geometric Numerical Methods for Robot
Simulation, Control and Optimization
Olivier Brüls
Department of Aerospace and Mechanical Engineering
University of Liège, Belgium
Advances in Numerical Modelling Newton Gateway to Mathematics, Isaac Newton Institute
Cambridge, December 3, 2019
Motivation for robot modelling
Low-level problems model-based methods
Direct and inverse kinematics
Local trajectory planning
Direct and inverse dynamics
Feedforward and feedback motion control
Force control
Collision detectionq1
q2
High-level problems machine learning methods
Global task planning, interactions with
complex and unstructured environments
Extension from rigid to flexible robot models?
p 3
Lightweight robots and machines
FlexPicker (KULeuven) 20-g robot (LIRMM)
Ralf (Georgia Tech)
Ella (Linz Center of Mechatronics) Sawyer collaborative robot
p 4
Why flexibility cannot be neglected?
Rigid models are only valid if dynamic excitations are well separated
from vibration eigenfrequencies
Tendency towards lightweight robots
Lower construction costs
Smaller actuators & lower energy consumption
Reduced overall bulkiness
Faster motions
Improved safety
Compliance is sometimes a desirable property
Indirect force limitation/control
Integration of sensing and actuation functions (soft-robotics)
Improved safety in human-robot cooperation tasks
p 5
What about state-of-the-art FEM tools?Simcenter/Samcef (Siemens) – MBS/FE formulation by (Géradin & Cardona 2001)
Rigid body /
FE mesh /
Superelement
Beams /
Superelements
Gear pairs
p 6
What about state-of-the-art FEM tools?
« Parameterize then discretize » paradigm
Pros:
Versatility and industrial maturity (GUI, CAD integration…)
Compatible with usual space & time discretization methods
Applicability to kinematic and dynamic problems in (flexible) robotics?
Stronger focus on the system intrinsic properties
Higher complexity and sensitivity of inverse problems
Cons:
Difficulty to preserve key properties of the solution
Numerical tricks to fix objectivity and locking issues
Parameterization-induced nonlinearities
Mostly limited to direct simulation problems
Alternative solutions based on differential geometry concepts?
p 7
First steps in differential geometry
First exposure during my PhD (2002-2005)
Differential geometry looks beautiful but out of my background?
Self-learning in 2006 with some influencial authors
Structural mechanics: Simo, Géradin, Cardona, Bottasso, Borri…
Control and robotics: Isidori, Sastry, Bullo, Murray, Zeitz, Selig…
Mathematics: Boothby, Crouch, Grossman, Munthe-Kaas, Iserles,
Celledoni, Owren…
Personal contributions since 2008
Tight encouragements by several colleagues in the community
Growing interest of the engineering community…
… yet the « abstraction wall » is still there
p 8
Development strategy
Revisit the FE approach using a differential geometry viewpoint
Represent finite motions as frame transformations
Consider these frame transformations as elements of the special
Euclidean group SE(3) (rotations and translations are coupled)
Local frames
Inertial frame
Represent the dynamics in the local (material) frame
reverse Copernician strategy
Exploit modern numerical methods on manifolds and Lie groups
elimination of parameterization nonlinearities and singularities
p 9
Outline
General introduction
Local frame modelling approach
Inverse dynamics & structural optimization
Conclusion
p 10
Rotation group - top motion example
O
ODE on a Lie group
The rotation matrix evolves on the Lie group SO(3)
The angular velocity matrix evolves on the Lie algebra so(3)
Lie group Euler explicit method
Special Euclidean group SE(3)
The group of rigid body motions SE(3) is represented by the set of 4x4
matrices
with and
Velocity on SE(3):
with
Velocities are thus naturally represented in the local frame
Dynamics in the local frame
Newton-Euler equations of a free rigid body
Representation in the local frame
with
Geometrically exact beam theory
Displacement gradient in local frame:
The strain operator is obtained as:
Linear relation!!!
Timoshenko-type nonlinear model
Paper by Simo & Vu-quoc (1985) cited almost 2000 times
Main difficulty: nonlinearity of the strain operator
Beam finite element interpolation
Discrete gradient
s = 0
s = L0
?
The shape functions define a helicoidal interpolation
These shape functions are geodesic curves on the group
By construction, the interpolation is frame invariant
Strains only depend on the relative motion:
Nonlinearities are reduced by mesh refinement!!!
Lie group interpolation:
(Sonneville, Cardona & B. 2014)
Example: dynamic beam
Generalized-a Lie group time integrator (B, Cardona & Arnold 2012)
This problem was solved without updating the iteration matrix in
the Newton iterations
No shear locking
p 16
Example: static beam roll-up
360° roll-up of a clamped beam [Simo & Fox 1989]
Poisson ratio: n = 0
Pure bending situation
Static solution: constant curvature
Numerical model 1
2 quadrangular elements
is updated at each Newton iteration
Exact solution in 1 load step
No shear locking (without any numerical trick)
Numerical model 2
2 quadrangular elements
is not updated
Exact solution in 2 load steps
Numerical model 3
4 quadrangular elements
is not updated
Exact solution in 1 load step
p 17
Pre-twisted beam
No shear/membrane
locking
Example: static shell response
p 18
Outline
General introduction
Local frame modelling approach
Inverse dynamics & structural optimization
Conclusion
p 19
Inverse dynamics of flexible robots
Inputs u
Outputs y
Flexible systems are underactuated
Direct dynamics: u(t) y(t) = ?
Inverse dynamics: y(t) u(t) = ?
p 20
Inverse dynamics: formulation
subject to
Numerical solution
Direct transcription on Lie group (Lismonde, Sonneville & B. 2019)
Gradient-based optimization
For systems with an unstable yet hyperbolic internal dynamics, the
inverse dynamics can be obtained by optimization
Kinematic + servo constraints
Multipliers + control forces
p 21
Inverse dynamics of a parallel robot
• Parallel robot with 3 rigid dofs.
• Made up of 2 tubular links (1/10 thick):
1. Rigid links (3):
Alu, 0.25 x 0.05 x 0.05 m.
2. Flexible links (3 x 4 beam elems):
Alu, 0.51 x 0.075 x 0.0075 m.
• Point mass at the end-effector (0.1 kg).
• Trajectory: half-circle with 0.1 m radius
in the xy plane, to be completed in 0.6 s.
p 22
Inverse dynamics of a parallel robot
Actual output trajectories before
and after optimization
Joint motions before and
after optimization
Nonlinearities mostly present in the joints (despite the finite motions)
Convergence achieved in 5-10 iterations
p 23
Experimental case: Sawyer
(Kang, Park & Arora, 2005)
• Imposed motion at revolute joints
• 2 beam elements per arm
• 4 design variables beam diameter
• Lumped mass at point A and at the tip
Structural optimization of a 2-dofs robot(Tromme et al, 2018)
(Kang, Park & Arora, 2005)
Multi-component constraint
Structural optimization of a 2-dofs robot(Tromme et al, 2018)
Structural optimization of a 2-dofs robot(Tromme et al, 2018)
p 27
Conclusions
Summary
Flexibility has a growing importance in robot modelling
State-of-the-art methods inappropriate for control & optimization
Local frame formulation + Lie group methods
Simplified and more intrinsic FE method
No locking or objectivity issue
Geometric nonlinearities reduced by mesh refinement
Solution to inverse dynamics and optimization problems
Perspectives
Real time performance for model-based control (model reduction)
Contact modelling (unilateral constraint with friction)
Grasping flexible objects
Human-robot collaboration
Virtual experiments for machine learning algorithms
Geometric Numerical Methods for Robot Simulation,
Control and Optimization
Olivier Brüls
Thanks for your attention!
Advances in Numerical Modelling Newton Gateway to MathematicsIsaac Newton Institute
Cambridge, December 3, 2019
