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An end effector is the device at a robot arm's working end that directly performs its task: for example, a gripper, welding tool or suction cup. The arm places and orients it, while the tool's geometry and contact behavior determine how that motion affects the object. A tool-tip frame can differ from the mounting-flange frame, so its offset must be included in forward kinematics.^[ref]

For example, suppose a two-finger gripper holds an object vertically by dry friction. If each finger applies normal force $N$ and the coefficient of friction is $\mu$, the ideal upward friction capacity is $2\mu N$. With $\mu =0.5$ and $N=10\mathrm{N}$ per finger, that capacity is $10\mathrm{N}$, just above the weight of a one-kilogram object near Earth's surface. A real test varies gripping force and load while observing slip; contact material, deformation and acceleration can invalidate the simple estimate. Positioning accuracy alone therefore does not establish grasp reliability.

Forward kinematics computes an end effector's configuration from a mechanism's joint coordinates and geometric model. For a serial robot arm, each joint contributes a relative displacement or rotation, and composing these transformations gives the tool's position and orientation relative to the base. This is a geometric calculation; the forces needed to realize the motion belong to dynamics.^[ref]

For a two-link planar arm, with relative joint angles ${\theta }_1,{\theta }_2$ and lengths $l_1,l_2$,For $l_1=l_2=1\mathrm{m}$, ${\theta }_1=\pi /2$ and ${\theta }_2=0$, the tool is at $(0,2)\mathrm{m}$ and points at angle $\pi /2$. Using ${\theta }_2$ alone for the second link's absolute angle would give the wrong result. Comparing this predicted position with an external measurement tests the link lengths, zero-angle calibration and rigid-link approximation.

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 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.

Robotics studies the design, modelling and control of machines that interact with their surroundings through sensing and motion. A robot combines sensors, actuators, computation and a mechanical structure. A robot arm must relate joint motion to the position and orientation of its end effector; a mobile robot must also estimate where it is relative to its environment.^[ref]

Kinematics asks which motions a mechanism permits. Dynamics determines the forces and torques needed to produce them, while feedback control corrects deviations using measurements. A desired path is useful only when the mechanism can realize it within constraints such as motor effort, contact and obstacles. These connections make robotics a meeting point of mechanics, geometry, statistical inference and control theory.