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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.

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 ${\alpha }_{\max }$.

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.

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.

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 quadrupedal robot is a robot with four legs used for support and locomotion. Its gait determines which feet contact the ground and when; having four legs does not imply that all four feet remain planted. Leg geometry, contact sensing and control of forces connect a desired body motion to feasible ground reactions.

An original statics calculation explains one advantage and one limitation of multiple feet. Assume point contacts on a horizontal floor, vertical gravity only, no contact moments and no acceleration. For robot mass $m$ and upward foot forces $F_i\ge 0$, force and moment balance requireHere $(x_c,y_c)$ is the vertical projection of the center of mass. Dividing by $mg$ expresses this point as a convex combination of foot locations. It must lie in their convex hull, the support polygon. A point on its boundary has no positive geometric margin against tipping in the corresponding direction.

For example, feet at $(\pm 0.50,\pm 0.30)\mathrm{m}$ surround a centered mass projection. Remove the front-left contact $(0.50,0.30)\mathrm{m}$: the origin now lies on the diagonal edge of the remaining triangle. The balance equations load the opposite two feet at $mg/2$ each and leave the third unloaded. Merely retaining three contacts has not created a positive margin. Shifting the mass projection to $(-0.10,-0.10)\mathrm{m}$ places it strictly inside that triangle, permitting three positive vertical reactions.

Dynamic running and jumping require a different balance including acceleration and angular momentum. An airborne robot has no ground support polygon at all. In the 2015 MIT Cheetah demonstration, onboard lidar estimated obstacle size and distance; control adjusted the approach strides and planned motor forces for a running jump.^[ref] The apparatus demonstrates coordinated perception and dynamic contact control, rather than a static support rule.

For a controlled walking test, record foot-contact times and body motion while changing one planned contact at a time. The geometric criterion above checks only the stipulated static model. Friction, actuator limits, terrain inclination and motion can make a geometrically supported pose physically infeasible. Conversely, a dynamic maneuver can pass through a pose that could not be maintained at rest.

A robot is a programmed, actuated mechanism that performs physical tasks such as locomotion, manipulation or positioning with some degree of autonomy. Its control system belongs to the robot, alongside the mechanism. The IFR’s primary classification document reproduces this broad ISO definition and distinguishes it from the narrower definition of an industrial manipulator.^[ref]

A practical robot combines a physical mechanism, actuators, task instructions and, where required by the task, sensing and feedback control. An arm’s joint arrangement constrains the positions it can reach; a mobile mechanism must also sustain contact and balance. The degree of autonomy is a task-specific property rather than a guarantee of unrestricted reasoning or competence.

An original two-link planar arm illustrates the relationship between structure and task. For unit link lengths and joint angles $q_1,q_2$, measured with the second angle relative to the first link, forward kinematics givesAt $q_1=0$ and $q_2=\pi /2$, the endpoint is $(1,1)$; with both angles zero it is $(2,0)$. Inverse kinematics asks which joint values achieve a requested endpoint, while control must make the actual mechanism follow the intended motion.^[ref]

Reaching a geometric target is only one part of an experiment. Position error, repeatability, applied force, execution time and behavior after a disturbance measure different aspects of performance. An arm can have an accurate kinematic model yet deviate under load; a controller can compensate only within its sensing, actuation and timing limits.

The MIT Cheetah robot provides a contrasting locomotion example: perception, contact planning and force generation are coordinated while the mechanism is moving. Such examples connect the physical meaning of a robot with mechanics, geometry, computation and experimental verification of a defined task.

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.

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]

The stance phase of a robot leg is the portion of a gait during which its foot is in the intended ground contact. It permits force transfer between the robot and its surroundings. Contact can occur with little load, and contact alone does not imply that the foot remains fixed without slipping. The swing phase instead transfers the foot toward another contact.

In an ideal rigid, nonadhesive contact model, let $g$ be the foot's normal gap above the ground and $F_n$ its normal contact force. The conditionsexclude penetration and tensile ground reaction. An open gap requires zero normal reaction; a zero gap permits a compressive reaction but does not require it to be strictly positive. The primary contact model distinguishes these unilateral constraints from the force/torque equations for equilibrium.^[ref]

For an original level-ground example, take a $20\mathrm{kg}$ robot supported by two stance feet, neglecting other external vertical forces. Newton's second law giveswhere $g_0=9.81\mathrm{m}{\mathrm{s}}^{-2}$ is gravitational acceleration and $a_z$ is upward center of mass acceleration. With equal sharing and $a_z=0$, each foot carries $98.1\mathrm{N}$. With $a_z=2\mathrm{m}{\mathrm{s}}^{-2}$, each carries $118.1\mathrm{N}$. Equal sharing is an additional symmetry assumption, not a consequence of knowing there are two contacts.

A simple Coulomb friction model constrains tangential force by $|F_t|\le \mu F_n$. At zero vertical acceleration and $\mu =0.3$, the two feet's combined horizontal-force capacity is at most $0.3m{g}_0=58.86\mathrm{N}$ in this planar model. Accelerating the body horizontally at $4\mathrm{m}{\mathrm{s}}^{-2}$ would require $80\mathrm{N}$ and violate that bound. A stance-time schedule alone cannot remove this force limit. Individual contact geometry and torque balance can impose further constraints even when the combined bound passes.

For a periodic cycle, contact duration $T_s$ gives duty cycle $\beta =T_s/T$. A proposed test records stance timing and forces separately, using calibrated sensors under the contacts together with body-motion measurements. It checks load transfer and slipping as well as touch timing. A support polygon is useful in an explicitly static planar model; dynamic stance additionally requires the changing force and momentum balance.

Uncertainty in a rigid-body bounding radius concerns whether the radius used in a motion bound actually encloses the robot about its declared pivot. A nominal mesh radius can miss manufacturing tolerances, an attached tool or an uncertain reference-point location. The bounding radius for rigid-body clearance must bound the relevant actual occupied geometry, not only a convenient displayed shape.

Let modeled body $\hat{B}$ and modeled pivot $\hat{c}$ satisfy $\Vert \hat{p}-\hat{c}\Vert \le \hat{R}$ for every $\hat{p}\in \hat{B}$. Suppose every point $p$ of the actual body lies within ${ϵ}_B$ of some modeled point, and the actual pivot obeys $\Vert c-\hat{c}\Vert \le {ϵ}_c$. The triangle inequality givesThus a certified radius isThe one-sided geometric coverage assumption is sufficient; it does not require every modeled point to lie close to actual material. A conservative overfilled model is allowed.

For an original example, take $\hat{R}=0.500\mathrm{m}$, shape coverage error ${ϵ}_B=3\mathrm{mm}$ and pivot-location error ${ϵ}_c=2\mathrm{mm}$. Then $R_{\mathrm{cert}}=0.505\mathrm{m}$. A ten-degree rotation about the actual fixed pivot moves any actual point by at mostUsing the nominal radius gives only $87.16\mathrm{mm}$, missing about $0.872\mathrm{mm}$ of possible motion. The angular clearance bound for a rotating body translates that displacement allowance into a clearance bound. Geometry and rotation must be considered together in the continuous-path planning model.^[ref]

Pivot-location uncertainty in a rotational clearance bound is distinct from an uncertain body translation. If the chosen reference point also moves, its translation contributes separately to the occupied-point motion. A speed certificate becomes $v_{\max }+R_{\mathrm{cert}}{\omega }_{\max }$ when the stated translational and angular speed bounds are valid. Substituting a smaller radius about a different center without accounting for that center’s motion changes the problem.

If the geometry has an additional bounded scale factor $s\le 1+{ϵ}_s$ about the modeled pivot, a sufficient bound is $(1+{ϵ}_s)\hat{R}+{ϵ}_B+{ϵ}_c$, under the explicitly separated scale and residual-shape assumptions. With ${ϵ}_s=0.01$ in the example it becomes $0.510\mathrm{m}$. Unknown errors and repeated descriptions of the same geometric tolerance must not be combined as if they were independent extra effects.

A radius enlargement addresses the motion-rate bound. It does not by itself correct an inaccurate sampled distance between the nominal robot surface and an obstacle. That query still needs a suitable geometric allowance or a clearance certificate with bounded distance-query error. Likewise, hard enclosure tolerances are different from standard uncertainties; a root-sum-square of standard deviations is not automatically a guaranteed radius.

A proposed geometric check measures extreme points of the assembled robot, including attachments, relative to the same pivot used by the motion representation. Account for measurement error and verify coverage between sampled surface points using a geometric model or validated tolerance. Recompute the bound when the tool, pivot or assembly changes, and record the radius’s provenance alongside the trajectory certificate.