locomotion concepts - eth z · autonomous systems lab autonomous mobile robots roland siegwart,...
Post on 29-Jul-2020
8 Views
Preview:
TRANSCRIPT
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Slides courtesy of Marco Hutter
Roland Siegwart, Margarita Chli, Nick Lawrance
1
Locomotion Concepts
Spring 2020
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Week 2 overview
Locomotion Concepts - add ons
2
2
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
On ground
Locomotion Concepts - add ons 3
Locomotion Concepts: Principles Found in Nature
25/02/2020
Reasons:
• Muscles don’t support actuation
• Nutrient transfer across rotational joints
• Traction
• Difficulties with obstacles (roll-over)
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Nature came up with a multitude of locomotion concepts
Adaptation to environmental characteristics
Adaptation to the perceived environment (e.g. size)
Concepts found in nature
Difficult to imitate technically
Do not employ wheels
Sometimes imitate wheels (bipedal walking)
The smaller living creatures are, the more likely they fly
Most technical systems today use wheels or caterpillars
Legged locomotion is still mostly a research topic
25/02/2020Locomotion Concepts - add ons 4
Locomotion Concepts
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Locomotion
physical interaction between the vehicle and its environment.
Locomotion is concerned with interaction forces, and the
mechanisms and actuators that generate them.
The most important issues in locomotion are:
25/02/2020Locomotion Concepts - add ons 5
Characterization of locomotion concept
stability
number of contact points
center of gravity
static/dynamic stabilization
inclination of terrain
characteristics of contact
contact point or contact area
angle of contact
friction
type of environment
structure
medium (water, air, soft or hard
ground)
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Legged systems can
overcome many
obstacles that are
not reachable by
wheeled systems!
But it is quite hard
to achieve this since
many DOFs must be controlled in a coordinated way
the robot must see detailed elements of the terrain
6
Why legged robots?
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
number of actuators
structural complexity
control expense
energy efficient
terrain (flat ground, soft
ground, climbing..)
movement of the involved
masses
walking / running includes up
and down movement of COG
some extra losses
25/02/2020Locomotion Concepts - add ons 7
Walking or rolling?
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
The fewer legs the more complicated becomes locomotion
Stability with point contact – at least three legs are required for static stability
Stability with surface contact – at least one leg is required
During walking some (usually half) of the legs are lifted
thus losing stability?
For static walking at least 4 (or 6) legs are required
Animals usually move two legs at a time
Humans require more than a year to stand and then walk on two legs.
25/02/2020Locomotion Concepts - add ons 8
Mobile Robots with legs (walking machines)
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
A minimum of two DOF is required to move a leg forward
a lift and a swing motion.
Sliding-free motion in more than one direction not possible
Three DOF for each leg in most cases (e.g. pictured below)
4th DOF for the ankle joint
might improve walking and stability
additional joint (DOF) increases the complexity of the design and
especially of the locomotion control.
25/02/2020Locomotion Concepts - add ons 9
Number of Joints of Each Leg (DOF: degrees of freedom)
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
The gait is characterized as the distinct sequence of lift
and release events of the individual legs
it depends on the number of legs.
the number of possible events N for a walking machine with k legs
is:
For a biped walker (k=2) the number of possible events N
is:
For a robot with 6 legs (hexapod) N is already
25/02/2020Locomotion Concepts - add ons 10
The number of distinct event sequences (gaits)
! 12 kN
6123! 3! 12 kN
800'916'39! 11 N
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
With two legs (biped) one can have four different states
1) Both legs down
2) Right leg down, left leg up
3) Right leg up, left leg down
4) Both leg up
A distinct event sequence can be considered as a change from one state to
another and back.
So we have the following distinct event sequences (change
of states) for a biped:
25/02/2020Locomotion Concepts - add ons 11
The number of distinct event sequences for biped:
6! 12 kN
Leg down
Leg up
1 -> 2 -> 1
1 -> 3 -> 1
1 -> 4 -> 1
2 -> 3 -> 2
2 -> 4 -> 2
3 -> 4 -> 3
turning
on right leg
hopping
with two legs
hopping
left leg
walking
running
hopping
right legturning
on left leg
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Changeover
Walking (trot)
Bounding
25/02/2020Locomotion Concepts - add ons 12
Most Obvious Natural Gaits with 4 Legs are
Dynamic
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 13
Muybridge 1878
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 14
Most Obvious Gait with 6 Legs is Static
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 15
Legged Robotics
Learning from Nature
Bio-inspired motor control 2012
[Iida]
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 16
Fundamental principles behind locomotion
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Static walking can be represented by inverted pendulum
17
Static walking principles
Inverted Pendulum
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Biped walking mechanism
not too far from real rolling
rolling of a polygon with side length equal
to the length of the step
the smaller the step gets, the more the
polygon tends to a circle (wheel)
But…
rotating joint was not invented by nature
work against gravity is required
more detailed analysis follows later in this
presentation
25/02/2020Locomotion Concepts - add ons 18
Biped Walking
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Forward falling combined with passive leg swing
Storage of energy: potential kinetic in combination with low
friction
25/02/2020Locomotion Concepts - add ons 19
Case Study: Passive Dynamic Walker
C y
outu
be
mate
rial
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Static walking can be represented by inverted pendulum
Exploit this in so-called passive dynamic walkers
20
Static walking principles
Inverted Pendulum
Energetically very efficient
usedE m g h hCOT
m g d m g d d
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Efficiency = cmt = |mech. energy| / (weight x dist. traveled)
Locomotion Concepts - add ons 21
Efficiency Comparison
C J. Braun, University of Edinburgh, UK
25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Static walking can be represented by inverted pendulum
Exploit this in so-called passive dynamic walkers
Add small actuation to walk on flat ground
22
Static walking principles
Inverted Pendulum
Cornell Ranger
Total distance: 65.24 km
Total time: 30:49:02
Power: 16.0 W
COT: 0.28
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 23
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 24
Towards Efficient Dynamic Walking: Optimizing
Gaits
C Structure and motion laboratory University of London
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Biomechanical studies suggest SLIP models to describe
complex running behaviors
Simple step-length rule to adjust the velocity
25/02/2020Locomotion Concepts - add ons 25
Dynamic Locomotion
Spring-Loaded Inverted Pendulum (SLIP) model
[Dickinson, Farley, Full 2000] [Geyer 2006]
, ,
1
2
FB
F HC des st R HC des HC HCT k h r r r r
Constant speed Accelerate Decelerate[Raibert 1986]
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
McGuigan & Wilson, 2003 – J. Exp. Bio.
Spring loaded inverted pendulum (SLIP)
are robust against collisions
can better handle uncertainties
can temporarily store energy
reduce peak power
[Alexander 1988,1990,2002,2003]
26
Dynamic locomotion
Leg Structure
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Inverted pendulum does not fully explain GRF
Use SLIP model to better explain what happens in nature
27
From static to dynamic locomotion
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Early Raibert hoppers (MIT leg lab) [1983]
Pneumatic piston
Hydraulic leg “angle” orientation
28
Dynamic Locomotion
SLIP principles in robotics
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Static locomotion (IP) Dynamic Locomotion (SLIP)
How to determine stability?
How to control (actively stabilize)?
29
Understanding Locomotion
Summary
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Consider a simple one-dimensional dynamic system,
described by an ODE:
ሶ𝑥 = 𝑓 𝑡, 𝑥
𝑥 𝑡′ = 𝑥0 +න0
𝑡′
𝑓 𝑥, 𝑡 𝑑𝑡
Consider the set of periodic systems, where:
𝑓 𝑡 + 𝑇, 𝑥 = 𝑓 𝑡, 𝑥 ∀(𝑡, 𝑥)
A simple example:
ሶ𝑥 = 𝑓 𝑡, 𝑥
= −𝑥 + 2 cos 𝑡 (𝑇 = 2𝜋)
25/02/2020Locomotion Concepts - add ons 30
Periodic dynamic systems
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 31
Periodic dynamic systems
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
For our simple system:
if 𝑥0 = 1
𝑥 2𝜋 = 1 = 𝑥0
This is a stable limit cycle with a period of 2𝜋
What happens for other values of 𝑥0?
25/02/2020Locomotion Concepts - add ons 32
Periodic dynamic systems
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 33
Cyclic stability and Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
if 𝑥0 = 1.1
𝑥 2𝜋 = 1.000186
Not quite a stable cycle… but appears to be approaching
one
For this simple system, we can solve the ODE, and
calculate the expression for 𝑥 𝑡, 𝑥0 :
𝑥 𝑡, 𝑥0 = 𝑒−𝑡 𝑥0 − 1 + cos 𝑡 + sin(𝑡)
Then, as 𝑡 → ∞, the first term disappears, and:
𝑥 𝑡 ≈ cos 𝑡 + sin(𝑡)
25/02/2020Locomotion Concepts - add ons 34
Cyclic stability and Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 35
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
The Poincaré map is defined as the function that
maps the endpoints of successive cycles:
𝑃 𝑥0 = 𝑥 𝑇 = 𝑥1
A fixed point 𝑥∗ describes a stable cycle:
𝑃 𝑥∗ = 𝑥∗
For our example, with period T = 2𝜋:
𝑥 𝑡, 𝑥0 = 𝑒−𝑡 𝑥0 − 1 + cos 𝑡𝑃 𝑥0 = 𝑥 2𝜋, 𝑥0 = 𝑒−2𝜋 𝑥0 − 1 + 1
= 𝑒−2𝜋𝑥0 + 1 − 𝑒−2𝜋
25/02/2020Locomotion Concepts - add ons 36
Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 37
Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
We can consider the Poincaré map iteratively:
𝑥 (𝑘 + 1)𝑇 = 𝑃 𝑥 𝑘𝑇
𝑥 𝑘𝑇 = 𝑃 𝑃 …𝑃 𝑥0
Something about the evolution of the Poincaré map
determines stability…
Formally, the derivative of 𝑃(𝑥) determines stability:
𝑖𝑓 𝑃′ 𝑥0 < 1 then 𝑥 𝑡 is asymptotically stable
𝑃′ 𝑥0 > 1 then 𝑥 𝑡 is unstable
25/02/2020Locomotion Concepts - add ons 38
Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
For our example, with period 2𝜋:
𝑥 𝑡, 𝑥0 = 𝑒−𝑡 𝑥0 − 1 + cos 𝑡 + sin 𝑡
𝑃 𝑥0 = 𝑥 2𝜋, 𝑥0 = 𝑒−2𝜋 𝑥0 − 1 + 1
= (𝑒−2𝜋)𝑥0 + 1 − 𝑒−2𝜋
Thus, the Poincare map is a line with gradient:
𝑒−2𝜋 ≈ 1.87 × 10−3
25/02/2020Locomotion Concepts - add ons 39
Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 40
Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 41
Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 42
Poincaré maps
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 43
Poincaré maps
Sta
ble
(𝑃′𝑥0
<1)
Unsta
ble
(𝑃′𝑥0
>1)
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 44
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Determining the derivatives in higher dimensions often
gets a bit tricky…
Essentially linearize 𝜕𝑃 𝑎
𝜕𝑥0around a fixed point 𝑎, and
calculate the eigenvalues of the linearized solution. If
𝜆 > 1 for any of the eigenvalues, the point 𝑎 is unstable.
Often requires numerical integration (no analytic solution)
25/02/2020Locomotion Concepts - add ons 45
Poincaré maps in higher dimensions
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 46
Dynamic Locomotion
Stability Analysis
x
yq Point mass:
Equation of Motion:
flight
stance
0x
y gq
, ,spring f k lF
xspring
o
yspring
o
F
x m
y Fg
m
q
Discuss the edX
limit-cycle movie
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 47
Stability of Locomotion
Limit Cycle Analysis
x
yq Point mass:
Equation of Motion:
flight
stance
0x
y gq
, ,spring f k lF
xspring
o
yspring
o
F
x m
y Fg
m
q
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 48
Stability of Locomotion
Limit Cycle Analysis
x
yq Point mass:
System state
How does the state
propagate from one
apex to the next?
x
y
x
y
z
y
xz
analysis at
apex
kz 1kz
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Given the Poincaré mapping from one apex to the next
P includes differential equations that cannot be analytically solved!
Analyze periodic stability of a fixed point
How can we do this? Linearization…
49
Stability of Locomotion
Limit Cycle Analysis
1k kP z z
* *Pz z
1k k k
P
z z Φ z
z
?
* *
*
2* * * 2
1 1 2...k k k k k
P PP P
x x x xz
z z z z z z z zx x
0
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Find the linearization of the Poincaré map around fix-point
Choose , with a small h
Simulate until the next apex, starting from
Calculate
Do the same thing for
50
Stability of Locomotion
Numerical Limit Cycle Analysis
*
1
1
k k
k k
y yP
x x
x xx
1
0
k
k
k
yh
x
z
1 *
1
1
0
k
k
yP h
x
z
*
*
1
*
1
*
*
k
k
y y
P h
x x
h
x xx
0
1
k
k
k
yh
x
z
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
The previous evaluation results in a 2x2 matrix
Eigenvalue analysis:
System is energy conservative:
System can be
stable
unstable:
51
Stability of Locomotion
Numerical Limit Cycle Analysis
*
P
x xx
1 1
2 1
2 1
Locomotion Concepts - add ons 25/02/2020
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance 25/02/2020Locomotion Concepts - add ons 52
Analyzing stability through limit cycles
static gaits
"System is statically stable"
Does not fall if stopped
Poincaré Map
Fix-Point
Linearization of mapping
The system is stable iff:
1k kP x x
* *Px x
1i Φ1k k k
P
x x Φ x
x
[C. David Remy, 2011]
||
ASL Autonomous Systems Lab
Autonomous Mobile Robots
Roland Siegwart, Margarita Chli, Nick Lawrance
Week 2 overview
Locomotion Concepts - add ons
2
53
top related