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

derivatives for design and control with Jim and Simon

review: serial manipulator 𝒚 end effector position 𝑦 ∈ ℝ 2 𝜄 3 𝜄 1 𝜄 2 ⋮ motor angles 𝜾 = 𝜄 1 𝜄 𝐿

forward kinematics (FK) what is 𝒚 𝜾 ? i.e., given joint angles 𝜾 , what is the corresponding tip position 𝒚 ?  something like 𝒚 𝜾 = 𝑼 𝐿 𝑺 𝐿 ⋯ 𝑼 1 𝑺 1 𝑷 // some big analytic // expression with a bunch // of sin(𝜄 𝑗 ) ’s and cos(𝜄 𝑘 ) ’s

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

option 0: solve analytically

option 1: use optimization minimize a suitable objective

option 1a: derivative-free optimization requires no derivatives - when in doubt just use CMA-ES

option 1b: derivative-based optimization may require 1 derivative (gradient)… gradient descent may require 2 derivatives (gradient and Hessian)… Newton’s method or be somewhere in the middle… Gauss-Newton, L-BFGS

option 2: learn it 𝑂 build a large set of training data {(𝜾 𝑗 , 𝒚 𝑗 )} 𝑗=1 using forward kinematics, then train a deep net using tensor flow, and evaluate the deep net at 𝜾

option 3: invert kinematics using the real world

option 1b: derivative-based optimization 1 𝒚) 𝑈 (𝒚 𝒒 − ෥ 2 (𝒚 𝒒 − ෥ Say the objective is 𝑔 𝒚(𝒒) = 𝒚) 𝑒𝑔 𝜖𝑔 𝑒𝒚 𝑒𝒒 = gradient is // chain rule 𝜖𝒚 𝑒𝒒 𝜖𝑔 𝜖𝒚 = (𝒚 𝒒 − ෥ 𝒚) is trivial to compute 𝑒𝒚 and for a serial manipulator, 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 𝐹(𝒚, 𝒒) 𝒚 Still want to solve optimization problems of this form: min 𝒒 𝑔 𝒚 𝒒

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 𝜖𝑔 d𝒚 𝜖𝒒 = 𝜖𝒚 𝑒𝒒 • But how to compute these derivatives, 𝑒𝒚 𝑒simulation 𝑒𝒒 = ? 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 𝒆𝐹 𝒉 𝒚, 𝒒 = 𝛂 𝐲 E = 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 𝜖𝒉 𝑒𝒚 𝜖𝒉 𝑒𝒒 = 𝑒𝒒 + 𝜖𝒒 = 𝟏 // total derivative 𝜖𝒚 • If the Jacobian 𝛼 𝒚 𝒉 is non-singular, we have −1 𝜖𝒉 𝑒𝒚 𝜖𝒉 𝑒𝒒 = − 𝜖𝒚 𝜖𝒒 16

Sensitivity Analysis • Used in many applications to quantify the sensitivity of a solution with respect to parameters 𝑒𝒚 ( 𝑻 = 𝑒𝒒 is also called the sensitivity matrix) • Widely for shape optimization, topology optimization, control, etc. 17

option 1b: derivative-based optimization 1 𝒚) 𝑈 (𝒚 𝒒 − ෥ 2 (𝒚 𝒒 − ෥ Say the objective is 𝑔 𝒚(𝒒) = 𝒚) 𝑒𝑔 𝜖𝑔 𝑒𝒚 𝑒𝒒 = gradient is // chain rule 𝜖𝒚 𝑒𝒒 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... 𝒚 , what is the optimal control 𝒒 ∗ ? given a target pose ෥ 𝑔 𝒚(𝒒) = 1 𝒚) 𝑈 𝑹(𝒚 𝒒 − ෥ 2 (𝒚 𝒒 − ෥ 𝒚)

real-world robot ignals 𝒒 ∗ optim imal control sig user-specified target pose ෥ 𝒚