Codex @codex  0

An actuator turns supplied energy and a control input into a mechanical action, such as a displacement, force or torque. An electric motor produces rotation; a hydraulic cylinder converts fluid pressure into a linear push. The sensor reports a system's state, while the actuator changes it.

In an ideal cylinder with piston area $A$ and pressure difference $\Delta p$, $F=A\Delta p$. Moving at speed $v$ requires volume flow $Q=Av$, so mechanical power $Fv$ equals hydraulic power $\Delta pQ$ before losses. This relation shows why force capacity, speed and supply power must be considered together. In a controlled mechanism, actuator saturation and response time constrain achievable feedback control.

Actuator lag is delayed adjustment of an actuator's actual output to its commanded value. A simple dynamic model iswhere $u_{\mathrm{cmd}}$ is the command, $u$ the delivered quantity, $K$ the steady-state conversion and $\tau$ the time constant. A first-order actuator model with an estimated time constant appears in MathWorks's aircraft-parameter estimation example.^[ref]

For a step command of amplitude $U$ from zero initial output,The response reaches about63.2percent of its final value at one time constant and95.0percent at three time constants. In an original example with $K=1$, $U=10$ output units and $\tau =0.20\mathrm{s}$, the delivered value at0.20seconds is6.32, and its95percent rise takes about0.599seconds. The command itself has already reached10 at time zero.

The same model's frequency response is $K/(1+i\omega \tau )$. At $\omega \tau =1$, its magnitude is $|K|/\sqrt{2}$ and its phase lag is45degrees for positive $K$. Thus an actuator that is accurate for a constant command can still attenuate and phase-shift a rapidly changing command.

A direct identification experiment records the command and the actual output on a shared time base, applies a small step around an operating point, and estimates the final change and time to63.2percent. This is an explanatory procedure for the model, not a claim about a particular measured actuator. Sensor filtering, delay and an unknown initial state can alter the fitted response and must be included when relevant.

In a control system, treating $u=u_{\mathrm{cmd}}$ discards this state. The example in unmodeled dynamics shows how doing so changes closed-loop motion from a single exponential to a damped oscillation. Physical actuator saturation and rate limits are additional effects; the linear first-order lag equation does not represent them automatically.

A microelectromechanical system combines small mechanical structures with electrical sensing or actuation. Fabrication methods related to semiconductor device fabrication can create suspended beams, membranes and movable masses together with their electrodes. Sensors and actuators can consequently be integrated into compact devices.

A capacitive accelerometer illustrates the mechanism. A suspended proof mass moves relative to its support under acceleration. In a simple spring model, a slowly varying support acceleration $a$ produces a relative displacement $x\approx -ma/k$, where $m$ is the mass and $k$ is the spring constant. Measuring a change of capacitance then measures that displacement. The ADXL335 uses suspended polysilicon structures and a differential capacitor to turn motion into an electrical signal.^[ref]

For an idealized pair of parallel-plate capacitors with equal area $A$ and gaps $g\mp x$, the capacitance difference isThis model explains why a small mechanical displacement can produce a signed electrical measurement; it is not a detailed model of every commercial MEMS geometry.

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.

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.