Dynamics Kris Hauser I400/B659, Spring 2014. Agenda Ordinary differential equations Open and closed...
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Transcript of Dynamics Kris Hauser I400/B659, Spring 2014. Agenda Ordinary differential equations Open and closed...
![Page 1: Dynamics Kris Hauser I400/B659, Spring 2014. Agenda Ordinary differential equations Open and closed loop controls Integration of ordinary differential.](https://reader036.fdocuments.in/reader036/viewer/2022081519/56649f2a5503460f94c44cb1/html5/thumbnails/1.jpg)
DynamicsKris HauserI400/B659, Spring 2014
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Agenda• Ordinary differential equations• Open and closed loop controls• Integration of ordinary differential equations• Dynamics of a particle under force field• Rigid body dynamics
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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)
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Dynamic equations• x(t): state trajectory
• (a function from real numbers to vectors)• Uncontrolled dynamic equation:
• An ordinary differential equation (ODE)
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Dynamic equations• x(t): state trajectory
• (a function from real numbers to vectors)• Uncontrolled dynamic equation:
• An ordinary differential equation (ODE)• Example: point mass with gravity g
• Position is • Acceleration (f=ma) is
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Dynamic equations• x(t): state trajectory
• (a function from real numbers to vectors)• Uncontrolled dynamic equation:
• An ordinary differential equation (ODE)• Example: point mass with gravity g
• Position is • Acceleration (f=ma) is • Uh… how do we work with this?• Second-order differential equation
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From second-order ODEs to first-order ODEs• Let with • Then
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From second-order ODEs to first-order ODEs• Let with • Then
• Here G is the gravity vector • If p is d dimensional, x is 2d-dimensional
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From time-dependent to time-independent dynamics• If • Let • Then
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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)?
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Numerical integration of ODEs• If and x(0) are known, then given a step size h,
• gives an approximate trajectory for k 1• Provided f is smooth• Accuracy depends on h
• Known as Euler’s method
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Integration errors• Lower error with smaller step size• Consider system whose limit cycle is a circle
• 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
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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
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Controlled Dynamics -> 1st order time-independent ODE• Open loop case:
• If , then let y(t)=(x(t),t)
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Controlled Dynamics -> 1st order time-independent ODE• Open loop case:
• If , then let y(t)=(x(t),t)
• Closed loop case:• If , then let y(t)=(x(t),t)
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Controlled Dynamics -> 1st order time-independent ODE• Open loop case:
• If , then let y(t)=(x(t),t)
• Closed loop case:• If , then let y(t)=(x(t),t)
• How do we choose u? A subject for future classes
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DYNAMICS OF RIGID BODIES
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Rigid Body Dynamics
• The following can be derived from first principles using Newton’s laws + rigidity assumption
• Parameters• CM translation c(t)• CM velocity v(t)• Rotation R(t)• Angular velocity w(t)• Mass m, local inertia tensor HL
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Rigid body ordinary differential equations• We will express forces and torques in terms of terms of H (a
function of R), , and
• Rearrange…
• 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
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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
wv
T
wv
H 0 0 m I
1/2
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Kinetic energy derivatives
• Force (@CM)
• H = [w]H – H[w]• Torque t = = [w] H w + H
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Summary
Gyroscopic “force”
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Force off of COM
x
F
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Force off of COM
x
F
Consider infinitesimal virtual displacement generated by F. (we don’t know what this is, exactly)The virtual work performed by this displacement is FT
𝛿𝑥
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Generalized torque
f
Now consider the equivalent force f, torque τ at COM
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Generalized torque
f
Now consider the equivalent force f, torque τ at COMAnd an infinitesimal virtual displacement of R.B. coordinates
𝛿𝑥
𝛿𝑞
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Generalized torque
f𝛿𝑥
𝛿𝑞
Now consider the equivalent force f, torque τ at COMAnd an infinitesimal virtual displacement of R.B. coordinates Virtual work in configuration space is [fT,τT]
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Principle of virtual work
f𝛿𝑥
𝛿𝑞
[fT,τT] = FT
Since we have [fT,τT] = FT
F
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Principle of virtual work
f𝛿𝑥
𝛿𝑞
[fT,τT] = FT
Since we have [fT,τT] = FT
Since this holds no matter what is, we have [fT,τT] = FTJ(q),
Or JT(q) F =
F
fτ
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Next class• Feedback control
• Principles App J