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Inverse kinematics finds joint coordinates that place an end effector at a requested position or configuration. It reverses the mapping specified by forward kinematics, but need not have a unique solution: a target can be unreachable, attainable through several configurations, or attainable through a continuous family when the mechanism has redundant degrees of freedom.^[ref]

For a two-link planar robot arm, the law of cosines givesWith unit link lengths and target $(1,1)$, $\cos {\theta }_2=0$. Two position solutions are $({\theta }_1,{\theta }_2)=(0,\pi /2)$ and $(\pi /2,-\pi /2)$. They have different arm shapes and tool orientations. A target at $(3,0)$ is unreachable because it lies beyond the total link length of two. Joint limits, obstacles and a specified tool orientation can exclude solutions even when the position is geometrically reachable.

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 robot arm is an articulated mechanism whose joints move an end effector, such as a gripper or tool, relative to a base. Forward kinematics maps joint coordinates to the tool's position and orientation; inverse kinematics asks which joint coordinates realize a desired tool configuration. The mechanism's degrees of freedom and joint limits constrain which configurations are reachable.^[ref]

For a planar arm with two rotating joints and link lengths $l_1,l_2$, let ${\theta }_1$ measure the first link's angle from the horizontal and ${\theta }_2$ the second link's angle relative to the first. Its tip satisfiesWith both lengths $1\mathrm{m}$, angles $(0,\pi /2)$ put the tip at $(1,1)\mathrm{m}$. Angles $(\pi /2,-\pi /2)$ reach the same position with a different arm shape and tool orientation. The example explains why inverse kinematics can have multiple solutions and why specifying position alone differs from specifying a full configuration.

In a physical experiment, joint encoders report angles while an external camera measures the tip. Comparing measured and predicted positions tests the geometric model; load-dependent deviations reveal flexibility or calibration errors that ideal kinematics omits. Feedback control then uses measured deviations to adjust the actuators.

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.

Swing-foot clearance is the separation between a robot's swinging foot and the terrain or obstacles it must pass. It concerns the whole swing phase, not just the maximum foot height. For a finite foot, use the relevant lowest material surface or full collision geometry; the height of a chosen ankle coordinate can differ from actual clearance.

An original point-foot model isolates vertical geometry. Let a planned foot position be $(x(s),z(s))$ and terrain height be $H(x)$, with swing parameter $s\in [0,1]$. DefineGround contact at the beginning and end normally gives zero endpoint clearance, so a positive lower bound must be assigned to the obstacle-crossing part of the path rather than incorrectly demanded throughout the closed contact-to-contact interval.

Use the constructed pathSuppose a flat-topped obstacle of height $15\mathrm{mm}$ occupies the horizontal range corresponding to $s\in [1/4,3/4]$, namely $31.25$ to $168.75\mathrm{mm}$. With $h=40\mathrm{mm}$, the lowest foot height over this interval is $h/2=20\mathrm{mm}$, attained at its edges. The minimum nominal obstacle clearance is therefore only $5\mathrm{mm}$ even though peak foot height is $40\mathrm{mm}$.

A robust swing-foot clearance margin includes specified error bounds. If actual foot height can be $3\mathrm{mm}$ below its plan and actual obstacle height $4\mathrm{mm}$ above the model, the guaranteed clearance is at leastThus the nominally clear path permits collision within those bounds. Guaranteeing at least $5\mathrm{mm}$ above this obstacle under the same errors requiresThe bounds add because both deviations can reduce clearance at once; they are deterministic limits, not independent standard uncertainties to combine by root-sum-square.

Greater height changes the leg's motion requirements. For a fixed swing duration $T_w$, this path has maximum vertical speed $\pi h/T_w$ and accelerations proportional to $h/T_w^2$. Clearance, timing, inverse kinematics and actuator constraints must therefore be checked together. A terrain discontinuity, horizontal position error or non-point foot additionally requires the corresponding geometry rather than just this vertical formula.

The primary planning text treats clearance as a possible planning objective and distinguishes local geometric constraints from dynamic ones.^[ref] A proposed robot test measures the foot surface and obstacle geometry in a shared reference frame, records the trajectory throughout the crossing, and compares the minimum separation with the stated bounds. Observing a high midpoint alone does not test the minimum-clearance claim.

The swing phase of a robot leg is the interval in which its foot moves between ground contacts without providing the intended ground support. It complements the stance phase. In a periodic gait with period $T$ and contact duty cycle $\beta$, its duration is $(1-\beta )T$. The leg can still require substantial actuator effort: moving its own mass and controlling its trajectory are different from supporting the robot through ground contact.

An original planar foot path gives an explicit design example. Let $s=t/T_w\in [0,1]$ parameterize a swing of duration $T_w$, horizontal travel $L$ and desired swing-foot clearance $h$ over a flat reference plane. ChooseBoth components have zero endpoint velocity. At mid-swing the foot has moved halfway horizontally and reaches height $h$. The maximum horizontal and vertical speed magnitudes areFor $L=0.20\mathrm{m}$, $h=0.040\mathrm{m}$ and $T_w=0.40\mathrm{s}$, these are $0.75\mathrm{m}{\mathrm{s}}^{-1}$ and about $0.314\mathrm{m}{\mathrm{s}}^{-1}$. Halving the available swing time doubles the path's speed requirements and quadruples its acceleration requirements. A contact schedule can therefore be feasible in timing arithmetic but too demanding for the actual leg.

The path is only a foot-level specification. Inverse kinematics must find joint configurations that realize it, and joint motion, collision and actuator constraints must also hold. Zero endpoint velocity does not guarantee continuous acceleration when this segment is joined to another trajectory. An uncertain or uneven ground height changes the actual clearance and contact time.

The primary contact-planning treatment distinguishes choosing contacts from moving through configuration space between them.^[ref] A proposed robot test measures the foot path, the contact transition and joint response while changing swing duration or clearance. Calibrated sensors can distinguish premature ground contact from a trajectory-tracking error. The example above is an original motion prescription; it is not a measured gait or a complete stability controller.