trajectory optimizationmaxim/classes/robotplanning_grad/...- explicit integration to satisfy...

Trajectory Optimizationor “How I stopped worrying and started loving the solvers”

some slides borrowed from Matthew Kelly

Ramkumar Natarajan

So far in the class...

Graph SearchSampling-Based

Trajectory Optimization

What is Trajectory Optimization?

?force

No actuation

An Optimal Trajectory

minimize: objective functionsubject to: dynamics and constraints

control(force)

state

An Optimal Trajectory

minimize: objective functionsubject to: dynamics and constraints

control(force)

state

An Optimal Trajectory

minimize: objective functionsubject to: dynamics and constraints

control(force)

state

Basic Form

Transcription

Direct Transcription

Complexity

u(t)

t0

t1 t2t3 t4

vs

z(k)

Complexity

u(t)

t0

t1 t2t3 t4

vs

Fundamentally Scalable- Only discrete in time (ALWAYS 1D)- No discretization in space- No exponential complexity/curse of

dimensionality

Decision Variables

subject to

What’s weird about

this?

t0 =

Decision Variables

subject toWhy is x(t) a

decision variable when

you have dynamics?

t0 =

Single Shooting

But x(t) can be written as a function of x0, u(t), f(x, u)

Assume linear system dynamics

Single ShootingThe optimization can be rewritten with unrolled dynamics and NO dynamics constraint

But hold on…

There is an An - Poor numerics- Ill-conditioning- Hard to set up the solver- Can be very powerful

Direct Collocation - Transcription

Direct Collocation - Transcription

The collocation constraints are in integral form using “quadrature” method

Can also use derivative form using “finite differences”

Direct Collocation - Transcription

Direct Collocation - Transcription

Direct Collocation - Interpolation

Initializing Trajectory Optimization

Basic Idea- Solve easier problem- Construct series of easier problems

Constructing Easier Problem- Neglect dynamics- Simplify objective- Omit constraints

Direct Collocation

Variants- Single shooting- Multiple shooting- Differential Dynamic

Programming

Classification

Trajectory Optimization

Direct Transcription

Shooting methods- Simulation- Explicit integration to

satisfy dynamics- Decision variables are

controls and boundary states

Simultaneous methods- Function

approximation- Dynamics are

constraints- Decision variables are

both states & controls

Variants- Direct collocation- Orthogonal collocation- Pseudospectral

collocation

Comparison

Graph SearchSampling-Based

Trajectory Optimization

- Strong theoretical properties- Simple to use- Global methods- Suitable to low dimensions

- Simple to use- Global methods- Scales to high dimensions- Relies on sampling strategy

- Very powerful capabilities- Local methods- Easily scales to very high dimensions- No guarantees on convergence/quality

Courses in RI16-745: Dynamic Optimization16-748: Underactuated Robotics

Transcribers

Solvers

IPOPT SNOPT FMINCON