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Adaptive clearance sampling spends more collision detection queries on portions of a trajectory that are difficult to certify, instead of accepting a path merely because a fixed set of samples is clear. Its guarantee comes from a bound on motion between samples. A configuration space path must satisfy the geometric constraints continuously.^[ref]

Represent the body by a nonempty compact set and the fixed obstacles by a nonempty closed set. Let $d(t)$ be their nonnegative set distance, and suppose a known Lipschitz continuity bound gives $|d(t)-d(s)|\le L|t-s|$. On an interval of duration $h$ with endpoint clearances $d_0,d_1$, each time is at most $h/2$ from one endpoint. Hence an endpoint certificate for continuous collision clearance isIf this lower bound is strictly positive, the entire interval is clear. For a required clearance margin $d_{\mathrm{safe}}\ge 0$, require the bound to exceed $d_{\mathrm{safe}}$ instead. Otherwise query the midpoint and subdivide. A sharper certificate uses both endpoint cones; when $L>0$ and the distances satisfy the assumed bound, its minimum is $(d_0+d_1-Lh)/2$. The simpler bound is sufficient for the example below.

For a translating and rotating rigid body, a valid speed estimate is $L=v_{\max }+R{\omega }_{\max }$, where $R$ is a bounding radius for rigid-body clearance, provided these are bounds over the whole interval and the obstacles stay fixed. The angular clearance bound for a rotating body proves the rotational contribution. A distance-estimation error bounded by $\epsilon$ can be accommodated by replacing each reported endpoint distance by its lower bound ${\hat{d}}_i-\epsilon$.

In an original point-motion example, $q(t)=(vt,0.01\mathrm{m})$ with $v=1m/s$ and $0\le t\le 1\mathrm{s}$ passes the obstacle point $(0.5,0)\mathrm{m}$. Its exact clearance is $d(t)=\sqrt{(vt-0.5\mathrm{m}{)}^2+(0.01\mathrm{m}{)}^2}$ and $L=1m/s$ suffices. Require $d_{\mathrm{safe}}=0.005\mathrm{m}$. The initial endpoint certificate is positive but below this required margin, so more queries are needed. Recursive midpoint sampling certifies long intervals away from the obstacle and refines only near the closest approach. The stated midpoint rule uses 15 distance queries and certifies 14 intervals. The figure shows the exact function alongside the sampled certificate; the formula is known here only because the example is constructed.

Unresolved intervals in adaptive collision checking must remain unresolved if a chosen depth or time budget is exhausted: failure to certify positive clearance does not establish a collision. Conversely, a query with exact zero set distance establishes contact or intersection. A sampled positive distance at or below $d_{\mathrm{safe}}$ violates the requested margin but does not establish contact. With exact queries, a valid finite $L$ and a trajectory whose clearance has a positive lower bound, subdivision eventually certifies every interval for $d_{\mathrm{safe}}=0$; a positive requested margin requires a positive surplus above that margin. Without positive clearance, finite termination is not promised.

A rigid body can hit an obstacle while its reference point stays fixed. An angular clearance bound for a rotating body controls the motion of every body point, allowing a collision detection calculation to reason about orientations between sampled poses.

Let a compact set $B$ lie within a bounding radius for rigid-body clearance $R$ of a fixed pivot. For a planar rotation matrix $Q_{\theta }$, a point $p\in B$ moves by a chord of a circle:The same correspondence bounds the Hausdorff distance between the two body sets. For a fixed nonempty closed set of obstacles $O$, let $d(\theta )$ be the nonnegative set distance between $Q_{\theta }B$ and $O$. The triangle inequality, applied to each point pair and then to their infima, givesThis is an explicit Lipschitz continuity bound. It concerns a fixed obstacle and the specified rigid geometry, rather than a universal constant for every robot.

To cover every intermediate orientation with excursion at most $\alpha \ge 0$ from a sampled pose, use $2R\sin (\min \{\alpha ,\pi \}/2)$, or the simpler $R\alpha$. The cap matters because the sine is not increasing beyond $\pi /2$. With translation displacement bounded by $\Delta$, the triangle inequality gives the sufficient clearance reduction $\Delta +R\alpha$.

For an original example, a body with $R=0.40\mathrm{m}$ rotating by at most $5{}^{\circ }$ moves any point by at most $2R\sin (2.5{}^{\circ })=34.90\mathrm{mm}$. An exact sampled clearance of $50\mathrm{mm}$ leaves at least $15.10\mathrm{mm}$ throughout that angular excursion when the pivot and obstacles stay fixed.

The whole excursion must be bounded, not just the difference between endpoint orientations modulo $2\pi$. A full revolution has identical endpoint orientations but can sweep through an obstacle. Likewise, a unit half-length rod rotating from $-30{}^{\circ }$ to $+30{}^{\circ }$ about its center clears an obstacle point at $(1,0)$ by $0.5$ at both endpoints but touches it midway. Adaptive clearance sampling therefore uses bounds on the actual interpolated motion. Configuration-space orientation and clearance are central to the continuous-path planning problem.^[ref]

A bounding radius for rigid-body clearance bounds the distance of every point of a rigid body from the reference point used to describe its motion. It converts angular motion into a bound on point displacement, so collision detection can account for the whole body between sampled poses rather than only its reference point.

Let $B$ be a nonempty compact set in body coordinates and let $c$ be the chosen pivot in those coordinates. A valid radius satisfiesThe maximum exists by compactness and continuity of the Euclidean norm. A bound larger than $R_{\ast }$ is conservative; a smaller one does not cover the specified geometry. For a convex polytope with vertices $v_i$, it is sufficient and exact to use $R_{\ast }={\max }_i\Vert v_i-c\Vert$: expressing a point as a convex combination and applying the triangle inequality proves that no interior point is farther away than the farthest vertex.

For an original rectangular-body example, take side lengths $0.6\mathrm{m}$ and $0.8\mathrm{m}$. A pivot at the centre gives $R_{\ast }=\sqrt{0.3^2+0.4^2}=0.5\mathrm{m}$. A pivot at a corner instead gives $R_{\ast }=\sqrt{0.6^2+0.8^2}=1.0\mathrm{m}$. For a $5{}^{\circ }$ rotation about the respective fixed pivot, the angular clearance bound for a rotating body isThe smaller radius cannot be used for the corner-pivot motion unless the changed motion of the reference point is also included. A convenient geometric centre does not automatically remain stationary.

For a moving reference point with speed at most $v_{\max }$ and angular speed at most ${\omega }_{\max }$, a point-speed bound is $v_{\max }+R{\omega }_{\max }$. This supplies a Lipschitz continuity constant for clearance against fixed obstacles. The role of body geometry and orientation in configuration space is developed in the primary planning text.^[ref]

The modeled body must include attached tools and protrusions relevant to the query. Uncertainty in a rigid-body bounding radius can be accommodated with a conservative geometric enlargement. This radius controls motion around the declared reference point; it does not replace the separate required clearance margin for uncertain obstacle locations or tracking errors.

A clearance bound between trajectory samples converts finitely many separation checks into a guarantee over a continuous trajectory, using a justified bound on how quickly clearance can change. Clear endpoints alone are insufficient: a foot or another robot part can approach an obstacle and move away again between them. The primary planning formulation requires constraints along the entire path.^[ref]

Let nominal clearance $c(s)$ have Lipschitz constant $L$ on a checked interval $[s_i,s_{i+1}]$ of length $h$. If both endpoint clearances are at least $c_{\min }$, then every intermediate point is within $h/2$ of an endpoint. ConsequentlyThis follows directly from $c(s)\ge c(s_i)-L|s-s_i|$ using the nearer endpoint. A uniform additional worst-case geometry or execution allowance $e$ gives actual clearance at least $c_{\min }-Lh/2-e$.

A tighter interval certificate retains the two different endpoint values, $a=c(s_i)$ and $b=c(s_{i+1})$. Writing $u=s-s_i$ gives two lower cones,If $L>0$ and $|a-b|\le Lh$, the cones meet inside the interval. Their common height yieldsIf the endpoint difference exceeds $Lh$, the proposed Lipschitz bound and exact endpoint values are inconsistent; simply applying the formula does not repair that inconsistency. With $L=0$, clearance is constant.

For an original example, take $h=0.10\mathrm{s}$, $L=0.20\mathrm{m}{\mathrm{s}}^{-1}$, $a=18\mathrm{mm}$ and $b=26\mathrm{mm}$. The simpler endpoint-minimum bound gives $8\mathrm{mm}$; the two-cone bound gives $12\mathrm{mm}$. Subtracting a $4\mathrm{mm}$ execution allowance leaves a certified $8\mathrm{mm}$ actual gap, sufficient for a requested $5\mathrm{mm}$ margin. These are lower bounds, not assertions that the trajectory reaches their values.

A counterexample explains why the allowance is necessary. On $[0,1]$, $c(s)=1-4s(1-s)$ has endpoint values one but touches zero at the midpoint. Checking only those endpoints misses the contact. Its derivative is bounded in magnitude by four, so a sufficiently fine sample grid with that valid bound can expose the missing guarantee.

The constant must match the geometry and parameter units. For a point moving relative to a fixed closed obstacle set, the distance to that set changes by at most the point’s displacement: the triangle inequality gives $|d_O(x)-d_O(y)|\le \Vert x-y\Vert$. A speed bound therefore supplies a time-based clearance bound. A rotating body requires a bound on every occupied point, such as the angular clearance bound for a rotating body; terrain-height clearance across a discontinuous edge cannot use a finite slope bound across that jump.

Adaptive clearance sampling subdivides intervals that fail the chosen certificate, concentrating computation where the margin or speed bound is least favorable. Failure of a conservative certificate means the interval needs better bounds or further checking; it does not itself prove a collision. For a proposed robot-foot test, validate the motion and geometry error bounds in one coordinate frame and compare the certified interval minima with dense independent measurements. Use the robust swing-foot clearance margin allowances without counting the same uncertainty twice.

A clearance certificate with bounded distance-query error proves separation over a continuous motion interval when sampled distances are known only within stated error bounds. A positive returned value from collision detection is not by itself a lower bound on the true geometric distance. Approximation, floating-point error and model error must be included if they affect the intended claim.

Let true set distance $d(t)\ge 0$ satisfy $|d(t)-d(s)|\le L|t-s|$ on $[0,h]$. At endpoints, assume certified query errors $|d_i-{\hat{d}}_i|\le {ϵ}_i$. DefineProvided the query bounds are consistent with nonnegative distances, ${\ell }_i\le d_i\le u_i$. The endpoint lower bounds propagate intoThis extends the endpoint certificate for continuous collision clearance. The primary planning model requires feasibility along the whole continuous path; checking endpoint poses alone is insufficient.^[ref]

The minimum of this envelope must be computed on the actual interval. Its general closed form, including unequal lower bounds, isFor $L>0$ and $|{\ell }_1-{\ell }_0|\le Lh$, the two sloping bounds meet inside the interval and the average expression dominates the other nonzero candidates. If ${\ell }_1-{\ell }_0>Lh$, their intersection lies before the interval; the minimum is instead $\max (0,{\ell }_1-Lh)$. The opposite inequality exchanges the endpoints. For $L=0$, the formula reduces to $\max ({\ell }_0,{\ell }_1)$ because true clearance is constant.

The lower bounds themselves need not satisfy the consistency condition for exact endpoint distances. For example, ${\ell }_0=2\mathrm{mm}$, ${\ell }_1=20\mathrm{mm}$ and $Lh=10\mathrm{mm}$ permit a true clearance rising from $10$ to $20\mathrm{mm}$. Rejecting these lower bounds because their difference exceeds $Lh$ would wrongly treat them as exact observations. The general certificate gives $10\mathrm{mm}$; using only the intersecting-cone expression would give the weaker $6\mathrm{mm}$.

A second original example takes reported distances $12$ and $20\mathrm{mm}$, errors $1$ and $3\mathrm{mm}$, and $Lh=10\mathrm{mm}$. Then ${\ell }_0=11$ and ${\ell }_1=17\mathrm{mm}$, so $b=9\mathrm{mm}$. The error-free formula would have claimed $11\mathrm{mm}$. For a required strict margin above $10\mathrm{mm}$, the uncertain queries do not certify the interval. Tightening both errors to $0.5\mathrm{mm}$ raises the lower bound to $10.5\mathrm{mm}$ and supplies the requested certificate.

Compatibility of bounded endpoint distance queries also uses the upper bounds. The intervals must obey ${\ell }_0\le u_0$, ${\ell }_1\le u_1$, ${\ell }_0\le u_1+Lh$ and ${\ell }_1\le u_0+Lh$. A violated condition signals inconsistent assumptions. An upper distance bound at or below a required margin can establish a sampled margin violation; failure of a lower-bound certificate alone does not establish a collision.

A proposed checker records the query’s enclosure guarantee, units and geometric model, propagates the motion bound with the actual interpolation, and leaves uncertified intervals unresolved for further subdivision or stronger queries. A standard deviation is not a deterministic query enclosure. A certificate against a nominal obstacle model also needs a separate justified relation to the actual obstacle geometry before it can be interpreted physically.

Combined pivot and axis uncertainty in rotational clearance bounds the occupied-point error of a rotating robot when its modeled pivot and axis direction are both uncertain. Adding separate bounds is valid only after specifying the same physical points, coordinate frame and endpoint quantity. A rigid rotation is represented by a rotation matrix, obtainable from a unit axis and angle through Rodrigues rotation formula.^[ref]

Let the nominal initial point and pivot be $\hat{p},\hat{c}$, with $r=\hat{p}-\hat{c}$ and $\Vert r\Vert \le \hat{R}$. Let actual values be $p=\hat{p}+\delta p$ and $c=\hat{c}+e$, where $\Vert \delta p\Vert \le {ϵ}_p$ and $\Vert e\Vert \le {ϵ}_c$. The actual and nominal rotations $Q,\hat{Q}$ have the same known principal angle $\theta \in [0,\pi ]$, while their unit axis directions differ by at most ${\alpha }_{\max }$. Expanding the actual endpoint and subtracting the nominal one gives the exact identityThe three terms isolate initial-point error, pivot error and direction-induced rotation error without assigning one error twice.

An orthogonal matrix preserves length, and $\Vert I-Q{\Vert }_2=2\sin (\theta /2)$. The axis-direction result in rotation-axis direction uncertainty in clearance bounds givesThereforeThe radius in this expression is the nominal $\hat{R}$, because the initial-point perturbation has already been accounted for by $Q\delta p$. Replacing it by an inflated radius is safe but may count some uncertainty conservatively twice. The pivot term uses the actual $Q$, whose rotation angle is shared and known in this model.

For an original tool-motion example, take $\hat{R}=0.50\mathrm{m}$, $\theta =90^{\circ }$, ${\alpha }_{\max }=1^{\circ }$, ${ϵ}_c=1\mathrm{mm}$ and ${ϵ}_p=0.2\mathrm{mm}$. The respective allowances are approximately $0.200$, $1.414$ and $12.341\mathrm{mm}$, totaling $E\approx 13.955\mathrm{mm}$. If every nominal occupied point is at least $20\mathrm{mm}$ from a fixed, exactly known obstacle, the actual clearance is at least $20-E\approx 6.045\mathrm{mm}$, provided point correspondence and the enclosure hold for the entire body at that orientation.

That last inference follows from the triangle inequality for set distance. An additional obstacle-position allowance must be subtracted separately when the obstacle is uncertain. A final-orientation calculation alone does not establish a bound along the intervening trajectory; robot motion planning constrains the whole continuous path.^[ref]

Shared-frame cancellation in rotational clearance uncertainty explains why a joint model can be tighter: translating the robot, pivot and obstacle coordinate descriptions together changes none of their relative clearances. A physical pivot displacement with a fixed obstacle is different. A proposed rotary-stage experiment fits pivot and direction in one frame, withholds fiducial observations at intermediate angles, and checks measured point errors against the bound. Uncertain rotation angle or changing structural deformation requires a further model; neither is included silently in ${\alpha }_{\max }$.

Compatibility of bounded endpoint distance queries asks whether two uncertain distance readings could arise from any nonnegative clearance function obeying a stated motion bound. It tests consistency of the observations and assumptions; it does not itself prove that the entire motion is collision-free.

Suppose endpoint queries yield intervals $I_0=[{\ell }_0,u_0]$ and $I_1=[{\ell }_1,u_1]$ for the true set distances, with $0\le {\ell }_i\le u_i$. Let the interval duration be $h>0$ and clearance have Lipschitz continuity constant $L\ge 0$. Write $c=Lh$. Compatibility means there exist $a\in I_0$ and $b\in I_1$ satisfying $|a-b|\le c$. The exact condition isalong with the two intervals being nonempty and nonnegative. Equivalently, the gap between the two intervals, $\max (0,{\ell }_0-u_1,{\ell }_1-u_0)$, must not exceed $c$.

Necessity follows because $a\ge {\ell }_0$, $b\le u_1$ and $a-b\le c$, with the reversed argument for the other inequality. For sufficiency, overlapping intervals allow $a=b$ at any common point. If they are disjoint, choose their nearest endpoints; the boxed condition bounds that separation by $c$. Linear interpolation between the selected nonnegative endpoints has slope at most $L$, proving that at least one admissible continuous clearance function exists.

The feasible endpoint intervals can be tightened toThese are the exact projections of the feasible endpoint-pair set onto each coordinate. A large difference between the lower bounds alone is not a contradiction: lower bounds are not exact distances.

For an original example, take $I_0=[4,8]\mathrm{mm}$, $I_1=[15,20]\mathrm{mm}$ and $Lh=10\mathrm{mm}$. Their nearest endpoints differ by $7\mathrm{mm}$, so the data are compatible. Tightening gives $I_0^{\prime }=[5,8]\mathrm{mm}$ and $I_1^{\prime }=[15,18]\mathrm{mm}$. By contrast, $[1,3]$ and $[15,17]\mathrm{mm}$ have a $12\mathrm{mm}$ minimum gap and are incompatible with the same motion bound.

Incompatibility means that some query enclosure, motion bound, timing or model assumption is wrong. It is not evidence that a collision occurred. Compatibility also need not certify separation: exact endpoint distances of $5\mathrm{mm}$ over one second with $L=10mm/s$ permit both constant clearance and a V-shaped clearance touching zero halfway through. The clearance certificate with bounded distance-query error evaluates a lower envelope over the full interval to answer that additional question.

For $L=0$, compatibility reduces to overlap of the intervals because clearance is constant. With more query times, multi-sample consistency of distance-query intervals must preserve one common clearance model rather than checking unrelated pairs with incompatible bounds. The primary motion-planning formulation requires constraints along a continuous path, motivating these between-sample checks.^[ref]

A proposed checker retains query lower and upper bounds, the clock interval, geometry version and motion allowance. It should report inconsistent inputs separately from undecided clearance intervals and certified margin violations. Tight numerical arithmetic alone cannot turn an unsupported standard uncertainty into a deterministic distance enclosure.

A configuration-space obstacle is a set of robot configurations whose occupied geometry intersects an obstacle. It translates a collision question about an extended robot into an excluded region of configuration space. The excluded coordinates can include rotation and joint angles as well as position. Hauser's primary planning formulation describes feasibility using the robot's occupied workspace set at each configuration.^[ref]

For an original fixed-orientation translation model, let $R$ be the robot's shape relative to its reference point, $O$ the physical obstacle and $q$ the reference-point translation. Collision meansThis is equivalent to $q+r=o$ for some $r\in R$, $o\in O$, and thereforeThe Minkowski sum must use the reflected robot shape. When $R$ is symmetric about its reference point, reflection leaves it unchanged; for an asymmetric reference choice, omitting the minus sign changes the collision region.

A one-dimensional example makes the reflection visible. Let a robot occupy $R=[0,0.30]$ metres relative to its left endpoint, and let the obstacle be the point $O=\{1\}$ metre. The forbidden reference positions are $[0.70,1.00]$ metres. At $q=0.80$, the robot spans $[0.80,1.10]$ and hits the obstacle. Using $O\oplus R=[1.00,1.30]$ would wrongly miss this collision and exclude some clear positions.

For a centered circular footprint of radius $r$ and a circular obstacle of radius $a$, the forbidden center positions form a disk of radius $a+r$. For axis-aligned rectangular shapes, each obstacle side expands by the robot's corresponding half-width. These are exact reductions for the stated geometry and fixed orientation.

An orientation-dependent configuration-space obstacle instead has a different translation-space cross-section for each orientation. A thin centered rod of length $0.40\mathrm{m}$ collides with a radius-$0.03\mathrm{m}$ obstacle centered at $(0.18,0)$ when the rod lies along the horizontal axis. At the same center position, a vertical rod is clear. The shared center point cannot encode the difference without the orientation coordinate.

For closed occupied sets, touching belongs to the forbidden set under this definition; a required clearance margin can enlarge it further. A proposed geometric check places the modeled robot at selected boundary configurations, compares the predicted intersections with measured geometry, and then verifies the path between them. Testing a few clear endpoint configurations alone does not establish a collision-free continuous motion.

A directional tipping margin is the distance an interior projection of the center of mass can move along one specified direction before reaching a boundary of its support polygon. It differs from the nearest-edge static tipping margin, which protects against movement in every horizontal direction. For a bounded polygon $P=\{x:a_i\cdot x\le b_i\}$, interior point $c$, and unit direction $u$, the ray $c+su$ stays inside exactly while all edge inequalities hold. HenceEdges with nonpositive denominator do not limit positive motion along that ray. The underlying gravity-only feasibility criterion is the nonnegative-reaction convex-hull condition in support polygon.^[ref]

For an original triangular footprint, take $x\ge 0$, $y\ge 0$, $x+y\le 1$ in metres and $c=(0.20,0.30)\mathrm{m}$. Moving right gives $s_{\max }=0.50\mathrm{m}$; moving downward gives $0.30\mathrm{m}$; moving along $u=(1,1)/\sqrt{2}$ gives $0.50/\sqrt{2}\approx 0.3536\mathrm{m}$. The all-direction margin is only $0.20\mathrm{m}$, set by the left edge. A displacement allowance restricted to the right can therefore be larger than the isotropic allowance without contradiction.

The same geometry can enter a planar load calculation. A horizontal force $Fu$ applied at height $h$ shifts the effective support-resultant point by $(Fh/mg)u$ under the corresponding static torque balance. The directional threshold is $F_{\mathrm{tip}}=mg{s}_{\max }/h$, subject also to friction and the other contact assumptions. With $m=10\mathrm{kg}$ and $h=0.50\mathrm{m}$, the rightward example gives $98.1\mathrm{N}$, whereas the nearest-edge worst-direction value is $39.24\mathrm{N}$.

A measured pull direction and contact layout are therefore part of the specification. Changes in contact set, applied couples, or slipping require the fuller analysis in tipping of a rigid body; this geometric directional reserve alone is not a dynamical stability certificate.

An endpoint certificate for continuous collision clearance uses distances at two sampled configurations and a bound on intervening motion to prove separation for an entire time interval. Its guarantee concerns every intermediate configuration, rather than only the sampled poses.

Let $d(t)\ge 0$ be the set distance between a moving nonempty compact set and fixed nonempty closed set of obstacles, with $0\le t\le h$. Suppose $|d(t)-d(s)|\le L|t-s|$, with $L\ge 0$, and let the exact endpoint distances be $d_0,d_1$. The two endpoint inequalities implyThis uses a valid clearance Lipschitz continuity constant, such as the rigid-body motion bound derived in angular clearance bound for a rotating body. The continuous-path requirement itself is fundamental to the primary planning model.^[ref]

For $L>0$ and consistent endpoint data $|d_1-d_0|\le Lh$, the two sloping bounds intersect atTheir maximum is minimized there, giving the sharp general lower boundIt is sharp among nonnegative functions with that Lipschitz continuity bound and those endpoints: the displayed envelope itself satisfies the assumptions. This does not say that the unknown actual trajectory attains the lower bound.

For an original example, take $d_0=12\mathrm{mm}$, $d_1=20\mathrm{mm}$, $L=100mm/s$ and $h=0.1\mathrm{s}$. Since $Lh=10\mathrm{mm}$, the simple nearest-endpoint certificate gives $\min (d_0,d_1)-Lh/2=7\mathrm{mm}$. The paired bounds give $(12+20-10)/2=11\mathrm{mm}$, with their intersection at $t_{\ast }=0.01\mathrm{s}$. Thus the sharper certificate proves separation greater than a $10\mathrm{mm}$ required clearance margin where the simpler one does not.

For $L=0$, consistency requires $d_0=d_1$, and the clearance is constant. An endpoint difference larger than $Lh$ contradicts the exact-data assumptions; the motion bound, query errors or model must be checked. A clearance certificate with bounded distance-query error replaces exact observations with justified lower bounds and must preserve the assumptions used in minimizing their envelope.

Under the strict convention of adaptive clearance sampling, certification requires the guaranteed lower bound to exceed the requested margin. If it does not, unresolved intervals in adaptive collision checking remain undecided until additional queries or other information resolve them. Failure of this sufficient certificate alone does not establish a collision.

Kris Hauser is a roboticist whose research concerns robot motion planning and control. The University of Illinois identifies him as a professor working across computing and engineering, with research on decision-making for physical robots under limits such as uncertain dynamics and bounded computation.^[ref]

His identified university profile provides an external portrait of Kris Hauser.

His author-hosted Robotic Systems by Kris Hauser brings geometry, motion selection and contact mechanics into one technical account. Its contact-planning treatment explains why selecting contact and selecting motion are coupled: making or breaking contact changes the constraints on the robot.^[ref] A collision-free trajectory alone does not establish that the available contacts can produce the required forces.

An original static planar example makes that distinction quantitative. Suppose two nonadhesive point contacts support a body of gravitational weight $W=200\mathrm{N}$ at horizontal positions $x_1=0$ and $x_2=0.5\mathrm{m}$. With center of mass coordinate $x_c$ and no other applied force or moment, vertical force and torque balance giveAt $x_c=0.20\mathrm{m}$, the normal reactions are $F_1=120\mathrm{N}$ and $F_2=80\mathrm{N}$. Moving the center to $x_c=0.60\mathrm{m}$ while retaining those same contacts would require $F_1=-40\mathrm{N}$ and $F_2=240\mathrm{N}$. A negative normal reaction violates the nonadhesive contact model. A planner must change the motion, contacts or assumed external forces; a path that avoids obstacles need not satisfy support constraints.

This is a derived example of the modeling question, not an experiment attributed to Hauser. His text supplies the explicit unilateral-contact and equilibrium framework.^[ref] A proposed robot test compares predicted reactions with calibrated contact measurements while shifting a controlled load, recording any change of contact rather than forcing the original static model onto a changed apparatus.

A multi-sample clearance envelope bounds every possible instantaneous clearance consistent with several uncertain distance queries and a known motion-rate limit. It sharpens isolated endpoint estimates by requiring one Lipschitz continuous clearance history to explain all samples. Such a bound addresses the between-sample gap in continuous robot motion planning, where valid sampled configurations alone do not guarantee a valid path.^[ref]

At times $t_i$, let $0\le {\ell }_i\le d(t_i)\le u_i$, and suppose the nonnegative clearance $d$ has Lipschitz constant $L$. Assuming multi-sample consistency of distance-query intervals, defineEvery admissible history obeys $d_{-}\le d\le d_{+}$. Each lower cone follows from $d(t)\ge d(t_i)-L|t-t_i|$; the upper cones follow by reversing the inequality. Under consistency, both envelope functions themselves satisfy the sample intervals and the same Lipschitz bound. Thus the enclosure is pointwise sharp among the scalar histories allowed by these assumptions.

A positive minimum of $d_{-}$ over a time interval certifies separation throughout it. If that minimum is zero, the assumptions permit a scalar history touching zero; this is lack of a strictly positive certificate, not an observation of collision. Some admissible scalar histories may not correspond to a realizable rigid-body path, so sharpness here concerns the stated scalar model.

The exact interval minimum is inexpensive to compute. On $[t_k,t_{k+1}]$, all sample times lie on one side or the other. PutThen $d_{-}(t)=\max (0,A_k-Lt,B_k+Lt)$. For $L>0$, its minimum occurs at the crossing $t_{\ast }=(A_k-B_k)/(2L)$ if that crossing lies in the interval; otherwise it occurs at the nearer endpoint. Prefix and suffix maxima therefore allow all segment minima to be computed in linear time. For $L=0$, every admissible clearance is constant and the lower envelope is ${\max }_i{\ell }_i$.

For an original example, queries at $0,1,2\mathrm{s}$ give intervals $[5,7]$, $[6,8]$, $[5,7]\mathrm{mm}$, and $L=4mm/s$. On the first segment, $A_0=5$ and $B_0=2$ in the corresponding millimetre/second coordinates. The crossing is $t_{\ast }=0.375\mathrm{s}$ and the lower bound there is $3.5\mathrm{mm}$. Symmetry gives a second minimum at $1.625\mathrm{s}$, so the full interval is certified with margin $3.5\mathrm{mm}$. Using only the two outer queries would give a minimum of $1\mathrm{mm}$.

A proposed checker first tests consistency, then computes segment minima with conservative arithmetic and reports the limiting time and active queries. Keep query errors, timestamps and the bound $L$ traceable to the same geometry and motion. Clearance envelope with segment-dependent speed bounds can use more motion information, but replacing $L|t-t_i|$ by an unsupported smaller number would invalidate the certificate.

Multi-sample consistency of distance-query intervals asks whether every bounded distance reading can belong to one nonnegative clearance function with the claimed Lipschitz constant. The same motion history must explain all samples. Checking each adjacent pair in isolation can select a different intermediate distance for each check and miss a contradiction.

Let strictly increasing times $t_0,\dots ,t_n$ have nonempty intervals $[{\ell }_i,u_i]$, with $0\le {\ell }_i\le u_i$, and let $L\ge 0$ bound clearance change per unit time. Necessary and sufficient conditions areNecessity follows by comparing any feasible distances $d(t_i),d(t_j)$ using Lipschitz continuity. For sufficiency, constructEach envelope has Lipschitz constant at most $L$. The pairwise conditions and the triangle inequality imply $f_{-}(t)\le f_{+}(t)$ everywhere. At each sample time, ${\ell }_i\le f_{-}(t_i)\le u_i$, so $f_{-}$ itself is an admissible nonnegative clearance history. This proves existence, not that this constructed history is the robot's actual history.

The intervals $[f_{-}(t_i),f_{+}(t_i)]$ are exact feasible projections at sample times. Both bounding functions are feasible and convex combinations of them remain feasible, so every intermediate value at one fixed time is attainable. The associated multi-sample clearance envelope also gives between-sample lower bounds for a clearance certificate with bounded distance-query error.

For an original counterexample to adjacent-only testing, let successive times differ by one second, take $L=4mm/s$, and useThe first adjacent pair can use middle distance $0$, and the second can use middle distance $10$; each pair is compatible. Yet the endpoints require a $10\mathrm{mm}$ change in two seconds, exceeding the allowed $8\mathrm{mm}$. No common history exists.

An equivalent linear-time feasibility test propagates a reachable interval. Start $J_0=I_0$ and recursively setStop if an interval becomes empty. In the example, $J_1=[0,4]$ and the next propagated upper bound is $8$, so $J_2$ is empty. Retaining the tightened interval is the key difference from independent adjacent-pair checks. If all intervals remain nonempty, select a final value and work backward through the reachable intervals; piecewise linear interpolation yields a feasible history.

Incompatibility diagnoses inconsistent readings or assumptions, not a demonstrated collision. Compatibility proves only the existence of a scalar clearance history; it does not prove that some rigid robot trajectory realizes that history or that the true clearance stays positive. Continuous collision avoidance remains a constraint along the physical path, as in the primary motion-planning formulation.^[ref] A proposed implementation tests synthetic histories with known bounds, deliberately inconsistent triples and timestamp errors, keeping model inconsistency separate from insufficient clearance.

An orientation-dependent configuration-space obstacle is the forbidden set for an extended robot whose occupied geometry changes with its orientation. For a planar rigid shape $B$ referred to a body-fixed point, translation $t\in {\mathbb{R}}^2$ and rotation matrix $R_{\theta }$, the occupied set is $t+R_{\theta }B$. With obstacle set $O$, collision meansFor each fixed angle, the forbidden translations are thereforeIndeed, an intersection gives $t+R_{\theta }b=o$, hence $t=o-R_{\theta }b$, and the converse follows by substitution. This is a reflected Minkowski sum, not an unreflected sum of the two shapes. Hauser develops these angle-indexed slices explicitly for a translating and rotating polygon.^[ref]

The full configuration-space obstacle is $\{(t,\theta ):t\in C_{\theta }\}$. Its angular coordinate is periodic: orientations zero and $2\pi$ represent the same rotation. Translation and orientation cannot generally be replaced by one planar obstacle inflation. A circular robot centered at its reference point is a special case, because rotating that shape leaves every $C_{\theta }$ unchanged.

An original doorway example uses a rectangle with half-length $a=0.30\mathrm{m}$ and half-width $b=0.10\mathrm{m}$, centered in an infinitely long horizontal corridor of width $W=0.25\mathrm{m}$. Its vertical half-extent at angle $\theta$ isWith contact counted as collision, a centered placement is clear only if $h(\theta )<W/2$. At zero angle the total width is $0.20\mathrm{m}$, leaving $25\mathrm{mm}$ at each wall. At $10^{\circ }$, the total width is about $0.3012\mathrm{m}$, so the same center position collides. For small positive angles, the first touching angle solves $0.30\sin \theta +0.10\cos \theta =0.125$, giving approximately $4.85^{\circ }$. A collision-free center path alone would miss this rotational restriction.

Angular sampling needs a bound on the geometry between checked orientations. If every body point lies within radius $\rho$ of the reference point, changing orientation by $\Delta \theta$ moves each corresponding point by at mostThe first expression is the chord length; the inequality follows from $\sin u\le u$. An angular clearance bound for a rotating body uses this motion allowance to transfer a measured separation to nearby orientations. For the rectangle above, $\rho =\sqrt{a^2+b^2}\approx 0.3162\mathrm{m}$. A ten-degree grid has a five-degree worst-case distance to a sampled orientation, allowing about $27.6\mathrm{mm}$ of point motion. A sampled $20\mathrm{mm}$ gap would not certify every intermediate orientation under that bound.

A geometric validation can place a rectangular test piece at predicted near-touching angles in a measured corridor, accounting for reference-point offsets and wall tolerances. A planner must also check the continuous motions connecting clear configurations; clear grid vertices alone do not establish clear connecting rotations. A conservative enclosing disk may simplify that check, at the cost of excluding some feasible narrow passages.

Pivot-location uncertainty in a rotational clearance bound describes how an error in the point about which a rigid rotation is modeled changes predicted occupied-point motion. A radius measured about one pivot is not automatically valid about another. The underlying planning problem must include both translation and orientation of the robot’s actual geometry.^[ref]

Let $Q$ be a fixed rotation matrix, $\hat{c}$ the modeled pivot, and $c=\hat{c}+e$ the actual pivot. First suppose the initial physical point $p$ is known exactly and both descriptions use the same rotation $Q$. The modeled and actual endpoints areso their difference is exactlyThis vanishes for zero rotation even if the pivot location is uncertain. Treating pivot error as an arbitrary full translation at every angle can therefore be unnecessarily conservative.

For a rotation through principal angle $0\le \theta \le \pi$ about a known axis direction, the component of $e$ along that direction is unchanged by $Q$. Its perpendicular component giveswhen $\Vert e\Vert \le {ϵ}_c$. The chord-length factor is the same geometry used in the angular clearance bound for a rotating body. For a $30^{\circ }$ rotation and ${ϵ}_c=2\mathrm{mm}$, the maximum endpoint error from pivot location alone is about $1.035\mathrm{mm}$. An equally large pivot offset parallel to the axis produces no endpoint change in this model.

If the initial point also has error $p-\hat{p}=\delta p$ with $\Vert \delta p\Vert \le {ϵ}_p$, thenFor the difference of displacement vectors, rather than final positions, the shape term is $(Q-I)\delta p$, giving the different bound $2({ϵ}_p+{ϵ}_c)\sin (\theta /2)$. The requested geometric quantity determines which allowance applies.

A complementary radius argument bounds total actual motion. If modeled points satisfy $\Vert \hat{p}-\hat{c}\Vert \le \hat{R}$, then $\Vert p-c\Vert \le \hat{R}+{ϵ}_p+{ϵ}_c$, and total point displacement is at most $2(\hat{R}+{ϵ}_p+{ϵ}_c)\sin (\theta /2)$. This is the enclosure approach in uncertainty in a rigid-body bounding radius; it need not equal an endpoint-error calculation against a nominal path.

Correlations can reduce the bound. If the same coordinate-frame translation shifts both the modeled point and pivot, relative geometry may be unchanged; treating those two descriptions as independent physical errors loses that cancellation. Conversely, rotation-axis direction uncertainty in clearance bounds changes $Q$ itself and is not covered by the fixed-$Q$ pivot formula.

A proposed rotary-stage check observes several rigid fiducials, estimates the pivot and axis in one coordinate frame, and compares independent endpoint measurements at several angles with the predicted chord scaling. Include attached tools and measurement uncertainty. When the robot moves against an uncertain obstacle, combine this occupied-point model with a separately justified distance-query enclosure; a correct pivot bound does not establish the obstacle model’s accuracy.

A required clearance margin is the minimum separation requested by a geometric motion specification. In adaptive clearance sampling, it is the threshold that a continuous-interval lower bound must exceed. Positive separation from an obstacle is enough to exclude contact in an exact model, but can still be less than the requested margin.

Let $d(t)$ be the modeled nonnegative set distance between a rigid body and fixed obstacles. Using the strict convention of adaptive clearance sampling, an interval is accepted only when its guaranteed lower bound exceeds $d_{\mathrm{safe}}\ge 0$. A query with $0<d(t)\le d_{\mathrm{safe}}$ violates that requested separation without demonstrating contact. Choosing $d_{\mathrm{safe}}$ is part of the specification, not something determined uniquely by the collision detection algorithm.

Geometric uncertainty can be incorporated with explicit bounds. Suppose robot geometry, obstacle geometry and reference-point tracking error can each change the relevant sets by at most ${\epsilon }_B,{\epsilon }_O,{\epsilon }_T$ in Hausdorff distance. The triangle inequality then givesTo guarantee an actual separation greater than $d_{\mathrm{req}}$, a sufficient modeled threshold isThese are simultaneous deterministic bounds. No independence assumption is needed to add them conservatively. The configuration-space construction in the primary planning text similarly represents body size by enlarging forbidden regions.^[ref]

For an original error budget, require $d_{\mathrm{req}}=5\mathrm{mm}$ and take geometric bounds of $2$, $3$ and $4\mathrm{mm}$ respectively. The modeled threshold becomes $14\mathrm{mm}$. Suppose the smaller reported endpoint distance is $20\mathrm{mm}$, the distance-query error is bounded by $1\mathrm{mm}$, the clearance Lipschitz continuity constant is $L=0.2m/s$, and the interval duration is $h=20\mathrm{ms}$. The simple interval certificate givesSubtracting the $9\mathrm{mm}$ geometric budget leaves at least $8\mathrm{mm}$ actual separation, exceeding the requested $5\mathrm{mm}$ by $3\mathrm{mm}$ in this model.

The query error was already subtracted from the interval bound and must not be subtracted again as an unrecorded duplicate allowance. Conversely, the motion term $Lh/2$ addresses unsampled motion; it does not automatically include obstacle-location error. An endpoint certificate for continuous collision clearance should state which effects are included in each quantity, while unresolved intervals in adaptive collision checking retain intervals whose lower bound is insufficient.

A roboticist develops or studies robots by combining their mechanical design, sensing, computation and control. The work connects a desired physical task to a machine that can actually perform it. Kinematics describes permitted motion; robot motion planning selects motions subject to constraints; feedback control uses measurements to correct execution. An algorithm and the apparatus it controls must therefore be evaluated together.

For an original planar robot arm example, let two links have lengths $l_1,l_2$ and relative joint angles ${\theta }_1,{\theta }_2$. Its end effector position isWith both lengths $0.5\mathrm{m}$ and angles $(0,\pi /2)$, the endpoint is $(0.5,0.5)\mathrm{m}$. A small first-joint offset $\delta {\theta }_1=0.01$ radians changes the position to first order by $(-y,x)\delta {\theta }_1=(-0.005,0.005)\mathrm{m}$, an error magnitude about $7.07\mathrm{mm}$. Accurate commanded numbers do not imply equally accurate physical geometry.

A proposed investigation compares measured endpoint positions with the kinematics model over several configurations, then checks whether the discrepancies follow a joint offset, a link-length error or changing mechanical deflection. Those alternatives suggest different corrections. It also tests execution on the real robot, rather than treating successful numerical planning as sufficient evidence of task completion.

Kris Hauser's author-hosted Robotic Systems by Kris Hauser connects geometric configurations, inverse kinematics, motion selection and contact constraints in one technical account.^[ref] Its contact discussion makes another key distinction: deciding when to touch the environment changes the applicable physical constraints. A roboticist may consequently work with mechanisms, algorithms or experiments while still needing a common model of what is sensed, commanded and physically achieved.

Robot motion planning computes a robot's proposed motion from its modeled state, environment and task constraints. A geometric planning problem asks for a path through configuration space; a timed motion can additionally constrain velocity, acceleration, forces and actuator effort. Hauser's primary text explicitly distinguishes geometric path planning from the broader problem involving dynamics and execution.^[ref]

Let $Q_{\mathrm{free}}$ be the configurations allowed by joint limits and collision detection. A geometric solution is a continuous curveEach point represents a complete robot configuration. A configuration-space obstacle is a forbidden set of configurations, not necessarily the same shape as the physical obstacle in the workspace.

An original planar example uses a point robot starting at $(0,0)$ and aiming for $(1,0)$, in metres. A circular obstacle centered at $(0.5,0)$ with radius $0.10\mathrm{m}$ blocks the straight path. A two-segment detour through $(0.5,0.20)$ has lengthThe perpendicular distance from the obstacle center to either segment isso the detour clears the circular obstacle. Checking only the start and goal would instead miss the collision on the straight segment.

Now give the robot a circular footprint of radius $0.05\mathrm{m}$, with translation as its only degree of freedom. The forbidden center positions form a disk of radius $0.15\mathrm{m}$: the obstacle is expanded by the footprint through Minkowski addition. The same detour still passes, but its center-path margin above the expanded obstacle is only about $0.0357\mathrm{m}$. The geometric reduction depends on the stated footprint and permitted motion.

A path's corner does not supply a dynamically smooth trajectory. Traversing this sharp detour at nonzero constant speed would instantaneously change velocity at its midpoint. A physical controller must stop, smooth the corner or choose a different feasible trajectory, while continuing to respect the obstacle and actuator constraints. A larger path length can be an acceptable cost if it permits the required motion.

A proposed validation compares the continuous planned geometry, sampled collision tests and measured execution on a controlled obstacle course. Report the robot geometry, constraints, computational budget and observed clearance. Failure to find a path in a limited run does not alone prove none exists; success on a numerical path does not alone establish accurate execution. Hauser treats completeness and the distinction between paths and feedback policies separately.^[ref]

A robust swing-foot clearance margin is a guaranteed lower separation between a robot's swing foot and terrain over a specified crossing interval and set of allowed errors. It differs from clearance computed only for a nominal trajectory. The interval must exclude intended ground contacts when a strictly positive margin is required.

An original point-foot model lets the planned terrain be $H(x)$ and the planned clearance be $c_0=z-H(x)$. Assume actual vertical foot error $|{\delta }_z|\le {ϵ}_z$, horizontal error $|{\delta }_x|\le {ϵ}_x$, and terrain-model height error bounded by ${ϵ}_H$. If $H$ has Lipschitz constant $L_H$ over the relevant neighborhood, thenThe actual clearance consequently satisfiesThis follows from the triangle inequality and covers all errors in the stated bounds. Their effects add in a worst-case calculation; combining them as independent random standard uncertainties would answer a different question.

For a local sloped-terrain example, take ${ϵ}_z=3\mathrm{mm}$, ${ϵ}_H=4\mathrm{mm}$, ${ϵ}_x=2\mathrm{mm}$ and $L_H=0.5$. The total clearance allowance is $3+4+0.5(2)=8\mathrm{mm}$. Guaranteeing a $5\mathrm{mm}$ margin therefore requires nominal clearance at least $13\mathrm{mm}$ throughout that crossing. A nominal $5\mathrm{mm}$ gap gives only a $-3\mathrm{mm}$ lower bound and cannot certify the requested separation.

Finite path samples need another allowance. Suppose the nominal clearance as a function of path parameter $s$ is Lipschitz with constant $L_c$, and every point lies within $\Delta s/2$ of a checked sample. If the smallest sampled clearance is $c_{\mathrm{sample}}$, a clearance bound between trajectory samples givesWith $L_c=0.10\mathrm{m}$ per unit parameter and sample spacing $\Delta s=0.10$, between-sample uncertainty costs another $5\mathrm{mm}$. The preceding example would then require sampled clearance at least $18\mathrm{mm}$ for a certified $5\mathrm{mm}$ margin.

Bounds must match the geometry. A vertical obstacle edge is not covered by a finite height-slope bound across its discontinuity; use the full configuration-space obstacle instead. Shared coordinate-frame errors can also cancel in a relative clearance, so separate component bounds can be conservative when their joint dependence is known.

The primary robotics formulation requires constraints along the full path and distinguishes planned from executed motion.^[ref] A proposed test records terrain and foot geometry in one reference frame, validates the stated error bounds, and compares a continuous-path certificate with observed minimum separation. A passing sample grid without the intervening bound does not establish the same guarantee.

Rotation-axis direction uncertainty in clearance bounds accounts for a robot rotating about a direction different from the modeled direction. Even an exactly located pivot does not determine the final occupied geometry if its axis direction is uncertain. Rotation acts through a rotation matrix; Rodrigues rotation formula computes that matrix from a unit axis and rotation angle.^[ref]

Assume the pivot $c$ and principal rotation angle $0\le \theta \le \pi$ are known and shared by both models. Let the unit axes be $u,\hat{u}$, separated by $\alpha$, and suppose every occupied point obeys $\Vert p-c\Vert \le R$. The endpoint discrepancy isIt measures error against a nominal endpoint, not total motion from the initial position.

One exact operator norm calculation uses the unit quaternions $(\cos (\theta /2),u\sin (\theta /2))$ and its modeled counterpart. Their scalar product isThe relative rotation has half-angle cosine $|d|$. A rotation through angle $\beta$ moves a perpendicular unit vector by $2\sin (\beta /2)$, givingConsequently, if the direction error is at most ${\alpha }_{\max }\le \pi$,The cap $2R$ follows because both endpoints lie on the same radius-$R$ sphere. The bound vanishes at zero rotation and, for small axis error, scales approximately as $2R\sin (\theta /2){\alpha }_{\max }$. Opposite oriented axes represent the same rotation at $\theta =\pi$; the conservative inequality need not capture that special cancellation.

For an original rotary-tool example, $R=0.50\mathrm{m}$, $\theta =90^{\circ }$ and ${\alpha }_{\max }=1^{\circ }$ give an endpoint-error allowance of $0.01234\mathrm{m}$, about $12.34\mathrm{mm}$. A direction tolerance that looks small numerically can consume a substantial clearance margin at a long tool tip.

A combined pivot and axis uncertainty in rotational clearance must add or jointly bound pivot-location error, angle error and direction error without double-counting a shared coordinate-frame error. The fixed-axis formula in pivot-location uncertainty in a rotational clearance bound addresses a different contribution. For a proposed test, observe multiple rigid fiducials over several angles, fit pivot and direction, and validate independent endpoint measurements against the claimed enclosure. Geometric robot motion planning needs the resulting occupied-body bound along the path, rather than only at its final orientation.^[ref]