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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 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.