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Algorithms and Numerical Methods for Motion Planning and

Motion Control: Dynamic Manipulation Assignments

Anton Shiriaev

Department of Engineering CyberneticsNorwegian University of Science and Technology

Trondheim, Norway([email protected])

April 21, 2016

Outline

Non-prehensile manipulation assignmentsExamples and featuresA “Butterfly” robot example

Steps in Motion PlanningChoices of CoordinatesSearching a Motion and a Motion Generator

Steps in Synthesis of ControllerTransverse Dynamics and Transverse Coordinates

Concluding remarks

Non-prehensile manipulation: examples and features

Non-prehensile manipulation is manipulation of an object

without a form or force-closure grasp


Dynamic non-prehensile manipulation offers:

◮ New robot primitives

◮ Flexibility in grasping objects of different shape and size

◮ Increased volume of reachable states in “robot+object” workspace






Benefits come at the expense of increased complexity in

◮ motion and trajectory planning

◮ motion control






Benefits come at the expense of increased complexity in

◮ motion and trajectory planning

◮ motion control

The challenges in developing algorithms are

◮ the need in developing dynamical models for a robot, an object and theirinteraction;

◮ under-actuation, i.e. when a number of degrees of freedom is larger than anumber of actuators;

◮ the presence of unilateral constraints

Outline




Concluding remarks

Conceptual scheme of a “Butterfly” robot

The tasks are

◮ to plan a rolling of a sphere on a frame

◮ to design a feedback controller to stabilize a motion

One of open challenges in robotics for two decades !!!

Outline




Concluding remarks

Choices of coordinates

~e10

~e20

~e21

~e11

~e12

~e22

y1

y2

x1 x2

θ2θ1

◮ Each body in plane requires 3 coordinates (x, y, θ) for describing itsconfiguration.

◮ The hand has a fixed point, therefore 4 quantities will be enough forreconstruction of the status of the hand and the disc


◮ Detecting of a contact point between the hand and the disc is challenging

◮ Contact models of rolling are simplistic


w~τ

~n

~e10

~e11

~e20

~e21

θ

~ρ(s)

s

~e12

~e22

ψ

R

B

A

◮ θ is the angle of rotation of the hand

◮ s is the distance to go to the shortest to the center of the disc point alongthe virtual curve γc

◮ w is the distance to that point along the normal direction

◮ ψ is the angle of rotation of the disc in the hand frame


w~τ

~n

~e10

~e11

~e20

~e21

θ

~ρ(s)

s

~e12

~e22

ψ

R

B

A

Properties valid for rolling without slipping

◮ w is zero, i.e. the hand and the disc has a point contact

◮d

dts = −R d

dtψ, i.e. the disc does not slip when rolls

Outline




Concluding remarks

Dynamics of the system

The dynamic model of the system in excessive coordinates q = (θ, ψ, s, w) is

Me(q)q̈ + Ce(q, q̇)q̇ +Ge(q) = [u, 0, 0, 0]T + F1 + F2

with u being a control variable; F1, F2 being the corresponding reaction forces.





The dynamics admit the reduction of the model for two variables (θ, ψ)

[

m11 m12m21 m22

] [

θ̈

ψ̈

]

+

[

c11 c12c21 c22

] [

θ̇

ψ̇

]

+

[

g1g2

]

=

[

u

0

]





The dynamics admit the reduction of the model for two variables (θ, ψ)

[

m11 m12m21 m22

] [

θ̈

ψ̈

]

+

[

c11 c12c21 c22

] [

θ̇

ψ̇

]

+

[

g1g2

]

=

[

u

0

]

We have to search for a forced periodic solution

θ∗(t), ψ∗(t), u∗(t)

of this system, for which the constraints hold.

Motion Generator

if the motion [θ∗(t), ψ∗(t)] is found then the disc will roll in one direction

⇓

the angle ψ∗(t) can be used for representation instead of time: θ∗(t) = Φ(ψ∗(t))

Motion Generator


⇓


⇓

d

dtθ∗ = d

dψΦ ddtψ∗, d

2

dt2θ∗ = d

dψΦ d

2

dt2ψ∗ + d

2

dψ2Φ

[

d

dtψ∗

]2

Motion Generator


⇓


⇓

d

dtθ∗ = d

dψΦ ddtψ∗, d

2

dt2θ∗ = d

dψΦ d

2

dt2ψ∗ + d

2

dψ2Φ

[

d

dtψ∗

]2

⇓

The passive dynamics for that motion

m21θ̈ +m22ψ̈ + c21θ̇ + c22ψ̇ + g2 = 0

can be re-written as a differential equation – Motion Generator – for ψ-variable

m21

[

Φ′ψ̈ + Φ′′ψ̇2]

+m22ψ̈ + c21Φ′

ψ̇ + c22ψ̇ + g2 = 0

[

α(ψ, {ki})ψ̈ + β(ψ, {ki})ψ̇2 + γ(ψ, {ki}) = 0

]

Motion Planning Algorithm

◮ Choose a set of synchronization functions: θ = Φ(ψ, k1, . . . , kn)

⇓

◮ Compute a family of Motion Generators parametrized by k1, . . . , kn

⇓

◮ Search for constants k∗1, . . . , k∗

n and one of the MG solution ψ∗(t) such

that the pair

ψ∗(t), θ∗(t) = Φ(ψ∗(t), k∗1, . . . , k

∗

n)

meets the unilateral constraint⇓

◮ If so, compute control variable u∗(t) from the system dynamics

m11θ̈∗ +m12ψ̈

∗ + c11θ̇∗ + c12ψ̇

∗ + g1 = u

Otherwise, modify the parameters and re-do the search.

Outline




Concluding remarks

Transverse Dynamics and Transverse Coordinates

S(0)TS(0)

S(t)

TS(t)

◮ Analysis of (in)stability of the nominal periodic trajectory and itsstabilization are defined by properties of the dynamics transverse to theorbit

◮ The system state vector has four components [θ, ψ, θ̇, ψ̇] ⇒ thedimension of the transverse dynamics is three.

◮ We need to define these quantities for [θ∗, ψ∗] and regulate them to zero

Important Remarks

◮ Non-prehensile manipulation tasks for robotics require combination of

◮ symbolic tools for modeling the dynamics and contact conditions

◮ identification and verification procedures for adjusting model parameters

◮ optimization procedures for model based trajectory planning

◮ control design methods and their stable numerical realization for orbitalstabilization

◮ Solving the task allows students practicing in integration of severaladvanced computational tools for approaching one of challengingreal-world problem in the lab

◮ Solving the task allows students learning advantages and weaknesses ofanalytical arguments, numerical procedures and their complementarity inapplications

◮ Safety, safety and safety versus entertainment and challenge in student labs

Non-prehensile manipulation assignmentsExamples and featuresA ``Butterfly'' robot example



Concluding remarks

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