Tangent Hyperplanes VI: Hybrid Systems
Why Smooth Tangent Spaces Are Not Enough
The first five parts assume that the dynamics are smooth over the interval being analyzed. That excludes many systems that matter in mechanics:
- foot strikes and ball impacts;
- contact making and breaking;
- friction-cone transitions;
- controllers that switch modes;
- guards that reset velocity or state.
Hybrid systems keep the tangent-space idea, but they no longer let one smooth Jacobian carry perturbations through the entire trajectory.
This compact part distills the rigorous version of Hybrid Tangent Spaces and preserves the critique that hybrid claims need explicit guard, reset, and Zeno assumptions.
Hybrid Automaton Model
A hybrid system can be represented by modes \(q \in Q\), continuous states \(x \in X_q\), vector fields \(f_q\), guard surfaces \(G_{ij}\), and reset maps \(R_{ij}\):
\[ \dot{x} = f_q(x,u), \qquad x^- \in G_{ij} \Rightarrow x^+ = R_{ij}(x^-). \]
Inside a fixed mode, the usual variational equation applies:
\[ \delta\dot{x} = A_q(t)\delta x + B_q(t)\delta u. \]
At a transition, the perturbation must be mapped through the event timing and reset. That map is the saltation matrix.
Saltation Matrix
For a guard with normal \(n = \nabla g(x^-)\), pre-impact vector field \(f^-\), post-impact vector field \(f^+\), and reset derivative \(D R\), a common saltation form is
\[ \Xi = D R + \frac{\left(f^+ - D R f^-\right)n^\top}{n^\top f^-}. \]
The denominator encodes transversality: the trajectory must cross the guard rather than slide tangent to it. If \(n^\top f^- \approx 0\), the saltation map is ill-conditioned and the tangent prediction is not trustworthy.
Within each smooth mode, the tangent dynamics are exact infinitesimally. Across a transverse event with a differentiable reset, the saltation matrix is the first-order map for perturbations through the event. The claim is local to that mode sequence and those regularity assumptions.
Mode-Aware Optimization
DDP and iLQR can be adapted to hybrid systems when the mode sequence is known or handled explicitly:
- propagate variational dynamics within each mode;
- apply the saltation matrix at each transition;
- run the backward pass through the composed linear maps;
- update the trajectory while checking whether event times or mode order changed.
If the mode order changes, the old linearization no longer describes the same piecewise-smooth object. The optimizer must either re-linearize under the new sequence or use a contact-implicit formulation that treats modes as decision variables.
Contact-Implicit Caution
Contact-implicit optimization can avoid preselecting a mode sequence, but it replaces the problem with complementarity constraints and numerical regularization choices. That is not a free lunch. It moves the difficulty from βwhich mode am I in?β to βhow accurately did the complementarity relaxation represent contact?β
Hybrid tangent-space language should not overstate novelty. Saltation matrices, impact mechanics, and contact-implicit optimization are established tools. The useful contribution here is editorial: connecting those tools back to the tangent-space reading path and making the scope assumptions visible.
Zeno and Grazing Events
Two edge cases deserve special treatment:
- Zeno behavior: infinitely many switches in finite time can make the composed tangent map undefined or physically irrelevant.
- Grazing contact: when the trajectory barely touches a guard, event timing is extremely sensitive and saltation terms can blow up.
A rendered page should not present a hybrid tangent calculation as reliable unless these cases are excluded or handled by the model.
Golf and Impact
Golf is a natural hybrid example because the club-ball collision is not a smooth continuation of the downswing dynamics. Tangent-space reasoning can still help, but the impact must be modeled as a mode transition with a reset or impulse map.
For pre-impact swing control, smooth variational dynamics may be adequate. For impact and launch, the model needs either an impact map or a separate collision model. Treating the whole motion as one smooth trajectory hides the most important discontinuity.
Reading Forward
Hybrid analysis explains when tangent maps jump. Part 7 uses residuals as runtime signals for when the current tangent model should be trusted, shrunk, updated, or abandoned.