Russ's Perspective

IK is a broad class of optimization problems, broadly asking to find "postures of the robot satisfying certain constraints".

(Then use a general solver to optimize.)

Greatest Hits of Optimization IK

Tommy's Perspective

IK is a coordinate system, where the pose of a robot is described by the position of its end effector(s), along with additional values to disambiguate kinematic redundancy.

Discrete

Continuous

Types of Kinematic Redunancy

Automatic Differentiation

Limited Applicability

• Analytic IK solutions are generally hand-designed.
• Some arms (booooo Kinova Jaco) don't have them yet.
• Solutions may not exist at all! (See: unsolvability of the quintic)
• The arm may be poorly calibrated

Setup

We have a robot with an analytic IK solution

\[\phi:\operatorname{SE}(3)\times\mathcal{C}\times\mathcal{D}\to\mathbb{R}^n.\]

where \(\mathcal{C}\) are continuous redundancy parameters and \(\mathcal{D}\) are discrete redundancy parameters.

What Does Analytic IK Look Like?

What Does Analytic IK Look Like?

Practicum: Domain Checking

• Joint limit violations
  • Check that \(\operatorname{IK}(X)\) is valid
• Non-reachable configurations
  • Check that \(\operatorname{FK}(\operatorname{IK}(X))\approx X\) (up to a tolerance)
• Inverse trig function domain
  • Clip inputs to the valid domain
  • Open question: what to do about gradients?
• Singular configurations
  • Ignore for now because they're a measure zero set.
  • Open question: can we do better?

Passively Avoiding Singularities?

Optimization IK (Standard Form)

\[\begin{array}{rl}\min_q & f(q)\\ \operatorname{subject\; to} & {}^WX^G=\operatorname{FK}(q),\\ & g_i(q)\ge0,\;\forall i\in I,\\ & h_j(q)=0,\;\forall j\in J.\end{array}\]

• \(f\) is our objective.
• \(g_i\) are generic inequality constraints.
• \(h_j\) are generic equality constraints.

Reformulated IK (Proposed)

\[\begin{array}{rl}\min_{X,\psi} & f(q)\\\operatorname{subject\; to} & T\in\operatorname{SE}(3),\psi\in\mathcal C,\\ & q:=\phi(X,\psi),\\ & \textrm{$q$ satisfies joint limits},\\ & {}^WX^G(q)=X,\\ & g_i(q)\ge0,\;\forall i\in I,\\ & h_j(q)=0,\;\forall j\in J.\end{array}\]

• \(q\) is not a variable, but is substituted
• "\(\textrm{$q$ satisfies joint limits}\)" is now a nonlinear inequality constraint
• The feasible set of \({}^WX^G(q)=X\) has positive measure.

Why it Might be Better

• Intuition: nonlinear equality constraints are inherently hard
  • You lose feasibility at every iteration
  • Constantly projecting back to feasibility is a costly slowdown
  • (This is why manifold optimization is so good.)
• Change-of-variables allows us to step along the manifold of solutions in closed form
• Possible problems
  • Anything that used \(q\) is now much more complicated
  • The new inequality constraints are complicated
  • It might be slower computationally

Some Practical Concerns

• Domain issues for the mappings
  • What if the initial guess is outside the domain?
  • What if an intermediate iterate leaves the domain?
  • (Is clipping inputs alone enough?)
• Discrete parameters
  • For now, just pick one and keep it constant
  • If varying initial guesses, also vary discrete parameters
• Representing rotations
  • Want to avoid nonlinear equality constraints
  • (Rotation matrices and quaternions are out)
  • Euler angles vs exponential coordinates

Collision-Free IK

\[\begin{array}{rl}\min_q & ||q||^2\\ \operatorname{subject\; to} & {}^WX^G=\operatorname{FK}(q),\\ & \operatorname{SignedDistance}(q)\ge 0.\end{array}\]

• Known to be a challenging problem
• Maybe easier if the optimizer doesn't have to worry about IK?

Humanoid Stability IK

\[\begin{array}{rl}\min_q & ||q||^2\\ \operatorname{subject\; to} & {}^WX^G_{\operatorname{left\;arm}}=\operatorname{FK}_{\operatorname{left\;arm}}(q),\\ & {}^WX^G_{\operatorname{right\;arm}}=\operatorname{FK}_{\operatorname{right\;arm}}(q),\\ & {}^WX^G_{\operatorname{left\;leg}}=\operatorname{FK}_{\operatorname{left\;leg}}(q),\\ & {}^WX^G_{\operatorname{right\;leg}}=\operatorname{FK}_{\operatorname{right\;leg}}(q),\\ & \pi_{xy}(x_{\operatorname{com}})\in\operatorname{SupportPolytope}(q).\end{array}\]

• We specify the pose of both hands and both feet.
• The support polytope is estimated as the convex hull of the corners of the feet.
• Could add gaze constraints.
• Could work with the spot robot as well.

The Part where Tommy Complains

Oh Great He's Still Complaining

Solution: Change the URDF

Multiple Possible EE Poses

\[\begin{array}{rl}\min_q & ||q||^2\\ \operatorname{subject\; to} & \operatorname{FK}(q)\in\{{}^WX^G:\textrm{some conditions}\}.\end{array}\]

Possible sets include

• Set of antipodal grasps on an object
  • Unclear how to formulate in an optimization problem
• Only fixing the position or orientation of the hand

Some Brainstorming Time...

• Create a generic \(n\)-link arm (where \(n\) can be scaled), and explore how performance changes as \(n\) grows.
• Solve IK problems for multiple robots jointly grasping an object.
• Footstep placement for a quadruped on rough terrain.
• Stewart platform
• ...