derivatives for design and control with Jim and Simon review: - - PowerPoint PPT Presentation

derivatives for design and control

with Jim and Simon

𝒚 𝜄2 𝜄1 𝜄3

motor angles 𝜾 = 𝜄1 ⋮ 𝜄𝐿 end effector position 𝑦 ∈ ℝ2

review: serial manipulator

forward kinematics (FK)

what is 𝒚 𝜾 ?

i.e., given joint angles 𝜾, what is the corresponding tip position 𝒚?

 something like 𝒚 𝜾 = 𝑼𝐿𝑺𝐿 ⋯ 𝑼1𝑺1𝑷 // some big analytic // expression with a bunch // of sin(𝜄𝑗)’s and cos(𝜄

𝑘)’s

inverse kinematics (IK)

what is 𝜾∗ ෥ 𝒚 ?

i.e., given joint target tip position ෥ 𝒚, what is an optimal choice of joint angles 𝜾∗?

• ption 0: solve analytically

• ption 1: use optimization

minimize a suitable objective

• ption 1a: derivative-free optimization

requires no derivatives

• when in doubt just use CMA-ES

• ption 1b: derivative-based optimization

may require 1 derivative (gradient)… gradient descent may require 2 derivatives (gradient and Hessian)… Newton’s method

• r be somewhere in the middle…

Gauss-Newton, L-BFGS

• ption 2: learn it

build a large set of training data {(𝜾𝑗, 𝒚𝑗)} 𝑗=1

𝑂

using forward kinematics, then train a deep net using tensor flow, and evaluate the deep net at 𝜾

• ption 3: invert kinematics using the real world

• ption 1b: derivative-based optimization

Say the objective is 𝑔 𝒚(𝒒) =

1 2 (𝒚 𝒒 − ෥

𝒚)𝑈(𝒚 𝒒 − ෥ 𝒚) gradient is

𝑒𝑔 𝑒𝒒 = 𝜖𝑔 𝜖𝒚 𝑒𝒚 𝑒𝒒

// chain rule

𝜖𝑔 𝜖𝒚 = (𝒚 𝒒 − ෥

𝒚) is trivial to compute and for a serial manipulator,

𝑒𝒚 𝑒𝒒 can be computed analytically

But what if 𝒚 𝒒 does not have an analytic expression?

For example, static equilibrium of a finite element mesh: 𝒚 𝒒 = 𝑏𝑠𝑕 min

𝒚

𝐹(𝒚, 𝒒) min

𝒒 𝑔 𝒚 𝒒

Still want to solve optimization problems of this form:

An example: topology optimization

Modeling continuous Relation between Parameters and State

• Observation: when we set parameters 𝒒, we observe the state 𝒚 as the result of simulation.
• Although 𝒚 are problem variables, they are not real DOFs – they are functions of the

parameters, i.e., 𝒚 = 𝒚(𝒒)

• Map from parameters to state is

𝒚 = simulate(𝒒)

• For design, we need derivatives of 𝒚(𝒒),

𝜖𝑔 𝜖𝒒 = 𝜖𝑔 𝜖𝒚 d𝒚 𝑒𝒒

• But how to compute these derivatives,

𝑒𝒚 𝑒𝒒 = 𝑒simulation 𝑒𝒒

?

• The derivative of an argmin...?

14

Differentiating the Map

• Although we can evaluate the map 𝒚 → 𝒚(𝒒), this map is not available in closed-form

(i.e., analytically)

• 𝒚 → 𝒚(𝒒) requires minimizing a function, i.e., solving a system of nonlinear equations.
• In general, it is impractical to compute derivatives of the minimization process.
• But even though 𝒚 → 𝒚(𝒒) is not given explicitly, the gradient of the objective

𝒉 𝒚, 𝒒 = 𝛂𝐲E =

𝒆𝐹 𝒆𝒚 = 𝟏

provides this map implicitly.

15

Differentiating the Map

• Suppose that (𝒚,𝒒) is a feasible pair, i.e., 𝒉(𝒚, 𝒒) = 𝟏. In other words, 𝒚 is an equilibrium

configuration for 𝒒.

• If we apply a parameter perturbation Δ𝒒, the system will undergo displacements Δ𝒚 such that

it is again in equilibrium, 𝒉(𝒚 + Δ𝒚, 𝒒 + Δ𝒒) = 𝟏

• Since this has to hold for arbitrary parameter variations, we have

𝑒𝒉 𝑒𝒒 = 𝜖𝒉 𝜖𝒚 𝑒𝒚 𝑒𝒒 + 𝜖𝒉 𝜖𝒒 = 𝟏 // total derivative

• If the Jacobian 𝛼

𝒚𝒉 is non-singular, we have 𝑒𝒚 𝑒𝒒 = − 𝜖𝒉 𝜖𝒚 −1 𝜖𝒉 𝜖𝒒

16

Sensitivity Analysis

• Used in many applications to quantify the sensitivity of a solution with respect to parameters

(𝑻 =

𝑒𝒚 𝑒𝒒 is also called the sensitivity matrix)

• Widely for shape optimization, topology optimization, control, etc.

17

• ption 1b: derivative-based optimization

Say the objective is 𝑔 𝒚(𝒒) =

1 2 (𝒚 𝒒 − ෥

𝒚)𝑈(𝒚 𝒒 − ෥ 𝒚) gradient is

𝑒𝑔 𝑒𝒒 = 𝜖𝑔 𝜖𝒚 𝑒𝒚 𝑒𝒒

// chain rule

𝜖𝑔 𝜖𝒚 = (𝒚 𝒒 − ෥

𝒚) is trivial to compute and for statically stable FEM (and for many, many other systems),

𝑒𝒚 𝑒𝒒 can be computed using sensitivity analysis

application: soft IK

say the control input 𝒒 are the contacted lengths of cables in a soft robot... given a target pose ෥

𝒚, what is the optimal control 𝒒∗?

𝑔 𝒚(𝒒) = 1 2 (𝒚 𝒒 − ෥ 𝒚)𝑈𝑹(𝒚 𝒒 − ෥ 𝒚)

real-world robot user-specified target pose ෥ 𝒚

• ptim

imal control sig ignals 𝒒∗