Transcript Non-Holonomic Motion Planning

Dynamics
Kris Hauser
I400/B659, Spring 2014
Agenda
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Ordinary differential equations
Open and closed loop controls
Integration of ordinary differential equations
Dynamics of a particle under force field
Rigid body dynamics
Dynamics
• How a system moves over time as a reaction to forces and
torques
• Distinguished from kinematics, which purely describes states
and geometric paths
• Uncontrolled dynamics:
• From initial conditions that include state x0 and time t0, the
system evolves to produce a trajectory x(t).
• Controlled dynamics:
• From initial conditions x0, time t0, and given controls u(t), the
system evolves to produce a trajectory x(t)
Dynamic equations
• x(t): state trajectory
• (a function from real numbers to vectors)
• Uncontrolled dynamic equation:
•
𝑑𝑥
𝑑𝑡
𝑡 = 𝑓(𝑥 𝑡 , 𝑡)
• An ordinary differential equation (ODE)
Dynamic equations
• x(t): state trajectory
• (a function from real numbers to vectors)
• Uncontrolled dynamic equation:
•
𝑑𝑥
𝑑𝑡
𝑡 = 𝑓(𝑥 𝑡 , 𝑡)
• An ordinary differential equation (ODE)
• Example: point mass with gravity g
• Position is 𝑝(𝑡) = 𝑝𝑥 (𝑡), 𝑝𝑦 (𝑡)
• Acceleration (f=ma) is
𝑑 2 𝑝𝑦 (𝑡)
𝑑𝑡 2
= −𝑔/𝑚
Dynamic equations
• x(t): state trajectory
• (a function from real numbers to vectors)
• Uncontrolled dynamic equation:
•
𝑑𝑥
𝑑𝑡
𝑡 = 𝑓(𝑥 𝑡 , 𝑡)
• An ordinary differential equation (ODE)
• Example: point mass with gravity g
• Position is 𝑝(𝑡) = 𝑝𝑥 (𝑡), 𝑝𝑦 (𝑡)
• Acceleration (f=ma) is
𝑑 2 𝑝𝑦 (𝑡)
𝑑𝑡 2
= −𝑔/𝑚
• Uh… how do we work with this?
• Second-order differential equation
From second-order ODEs to
first-order ODEs
• Let 𝑥(𝑡) = (𝑝(𝑡), 𝑣(𝑡)) with 𝑣(𝑡) =
• Then
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𝑑𝑥
𝑑𝑡
𝑡 =
𝑑𝑝
(𝑡)
𝑑𝑡
𝑑𝑣
(𝑡)
𝑑𝑡
𝑑𝑝 𝑡
𝑑𝑡
From second-order ODEs to
first-order ODEs
• Let 𝑥(𝑡) = (𝑝(𝑡), 𝑣(𝑡)) with 𝑣(𝑡) =
𝑑𝑝 𝑡
𝑑𝑡
• Then
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𝑑𝑥
𝑑𝑡
𝑡 =
𝑑𝑝
(𝑡)
𝑑𝑡
𝑑𝑣
(𝑡)
𝑑𝑡
=
𝑣 𝑡
𝐺
𝑚
= 𝑓 𝑥 𝑡 ,𝑡
0
−𝑔
• If p is d dimensional, x is 2d-dimensional
• Here G is the gravity vector
From time-dependent to timeindependent dynamics
• If
𝑑𝑥
(𝑡)
𝑑𝑡
= 𝑓(𝑥 𝑡 , 𝑡)
• Let y(𝑡) = (𝑥 𝑡 , 𝑡)
• Then
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𝑑𝑦
𝑑𝑡
𝑡 =
𝑑𝑥
(𝑡)
𝑑𝑡
𝑑𝑡
(𝑡)
𝑑𝑡
=
𝑓 𝑥 𝑡 ,𝑡
1
=𝑔 𝑦 𝑡
Dynamic equation as a vector
field
• Can ask:
• From some initial condition, on what trajectory does the state
evolve?
• Where will states from some set of initial conditions end up?
• Point (convergence), a cycle (limit cycle), or infinity (divergence)?
Numerical integration of ODEs
• If
h,
𝑑𝑥
(𝑡)
𝑑𝑡
= 𝑓(𝑥(𝑡)) and x(0) are known, then given a step size
• 𝑥 𝑘ℎ  𝑥𝑘 = 𝑥𝑘−1 + ℎ 𝑓(𝑥𝑘−1 )
• gives an approximate trajectory for k 1
• Provided f is smooth
• Accuracy depends on h
• Known as Euler’s method
Integration errors
• Lower error with smaller step size
• Consider system whose limit cycle is a circle
•
𝑑𝑥
𝑑𝑡
𝑑𝑦
𝑑𝑡
−𝑦
=
𝑥
• Euler integrator diverges for all step sizes!
• Better integration schemes are available
• (e.g., Runge-Kutta methods, implicit integration, adaptive step
sizes, energy conservation methods, etc)
• Beyond the scope of this course
Open vs. Closed loop
• Open loop control:
• The controls u(t) only depend on time, not x(t)
• E.g., a planned path, sent to the robot
• No ability to correct for unexpected errors
• Closed loop control :
• The controls u(x(t),t) depend both on time and x(t)
• Feedback control
• Requires the ability to measure x(t) (to some extent)
• In either case we have an ODE, once we have chosen the control
function
st
Controlled Dynamics -> 1
order time-independent ODE
• Open loop case:
• If
•
𝑑𝑥
(𝑡)
𝑑𝑡
𝑑𝑦
𝑑𝑡
= 𝑓(𝑥 𝑡 , 𝑢(𝑡)), then let y(t)=(x(t),t)
𝑡 = 𝑓 𝑥 𝑡 ,𝑢 𝑡
1
=𝑔 𝑦 𝑡
st
Controlled Dynamics -> 1
order time-independent ODE
• Open loop case:
• If
•
𝑑𝑥
(𝑡)
𝑑𝑡
= 𝑓(𝑥 𝑡 , 𝑢(𝑡)), then let y(t)=(x(t),t)
𝑡 = 𝑓 𝑥 𝑡 ,𝑢 𝑡
1
𝑑𝑦
𝑑𝑡
=𝑔 𝑦 𝑡
• Closed loop case:
𝑑𝑥
• If 𝑑𝑡 (𝑡) = 𝑓(𝑥 𝑡 , 𝑢(𝑥 𝑡 , 𝑡)), then let y(t)=(x(t),t)
•
𝑑𝑦
𝑑𝑡
𝑡 = 𝑓 𝑥 𝑡 ,𝑢 𝑥 𝑡 ,𝑡
1
=𝑔 𝑦 𝑡
st
Controlled Dynamics -> 1
order time-independent ODE
• Open loop case:
• If
•
𝑑𝑥
(𝑡)
𝑑𝑡
= 𝑓(𝑥 𝑡 , 𝑢(𝑡)), then let y(t)=(x(t),t)
𝑡 = 𝑓 𝑥 𝑡 ,𝑢 𝑡
1
𝑑𝑦
𝑑𝑡
=𝑔 𝑦 𝑡
• Closed loop case:
𝑑𝑥
• If 𝑑𝑡 (𝑡) = 𝑓(𝑥 𝑡 , 𝑢(𝑥 𝑡 , 𝑡)), then let y(t)=(x(t),t)
•
𝑑𝑦
𝑑𝑡
𝑡 = 𝑓 𝑥 𝑡 ,𝑢 𝑥 𝑡 ,𝑡
1
=𝑔 𝑦 𝑡
• How do we choose u? A subject for future classes
DYNAMICS OF RIGID BODIES
Rigid Body Dynamics
• The following can be derived from first principles
using Newton’s laws + rigidity assumption
• Parameters
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CM translation c(t)
CM velocity v(t)
Rotation R(t)
Angular velocity w(t)
Mass m, local inertia tensor HL
Rigid body ordinary
differential equations
• We will express forces and torques in terms of terms of H (a
function of R), 𝜔, 𝑣 and 𝜔
• 𝑓 = 𝑚𝑣
• 𝜏 = [𝜔] 𝐻 𝜔 + 𝐻 𝜔
• Rearrange…
• 𝑣 = 𝑓/𝑚
• 𝜔 = 𝐻 −1 (𝜏 − 𝜔 𝐻 𝜔 )
• So knowing f(t) and τ(t), we can derive c(t), v(t), R(t), and w(t)
by solving an ordinary differential equation (ODE)
• dx/dt = f(x)
• x(0) = x0
• With x=(c,v,R,w) the state of the rigid body
Kinetic energy for rigid body
• Rigid body with velocity v, angular velocity w
• KE = ½ (m vTv + wT H w)
• World-space inertia tensor H = R HL RT
1/2
w
v
T
H 0
0 mI
w
v
Kinetic energy derivatives
•
𝜕𝐾𝐸
𝜕𝑣
= 𝑚𝑣
• Force (@CM) 𝑓 =
•
•
𝜕𝐾𝐸
𝜕𝜔
𝑑
H
𝑑𝑡
𝑑 𝜕𝐾𝐸
( )
𝑑𝑡 𝜕𝑣
= 𝑚𝑣
= 𝐻𝜔
= [w]H – H[w]
• Torque t =
𝑑 𝜕𝐾𝐸
𝑑𝑡 𝜕𝜔
= [w] H w + H 𝜔
Summary
•𝑓 = 𝑚𝑣
• 𝜏 = [𝜔] 𝐻 𝜔 + 𝐻 𝜔
Gyroscopic “force”
Force off of COM
F
x
Force off of COM
F
𝛿𝑥
x
Consider infinitesimal virtual displacement 𝛿𝑥 generated
by F.
(we don’t know what this is, exactly)
The virtual work performed by this displacement is FT𝛿𝑥
Generalized torque
f
Now consider the equivalent force f, torque τ at COM
Generalized torque
f
𝛿𝑥
𝛿𝑞
Now consider the equivalent force f, torque τ at COM
And an infinitesimal virtual displacement of R.B.
coordinates 𝛿𝑞
Generalized torque
f
𝛿𝑥
𝛿𝑞
Now consider the equivalent force f, torque τ at COM
And an infinitesimal virtual displacement of R.B.
coordinates 𝛿𝑞
Virtual work in configuration space is [fT,τT] 𝛿𝑞
Principle of virtual work
f
F
𝛿𝑥
𝛿𝑞
[fT,τT] 𝛿𝑞= FT 𝛿𝑥
Since 𝛿𝑥 = 𝐽 𝑞 𝛿𝑞 we have [fT,τT] 𝛿𝑞= FT𝐽 𝑞 𝛿𝑞
Principle of virtual work
f
F
𝛿𝑥
𝛿𝑞
[fT,τT] 𝛿𝑞= FT 𝛿𝑥
Since 𝛿𝑥 = 𝐽 𝑞 𝛿𝑞 we have [fT,τT] 𝛿𝑞= FT𝐽 𝑞 𝛿𝑞
Since this holds no matter what 𝛿𝑞 is, we have [fT,τT] =
FTJ(q),
f
Or JT(q) F = τ
Next class
• Feedback control
• Principles App J