Forward Kinematics (2 DOF)
Compute the (x, y) position of the end of a planar 2-degree-of-freedom robotic arm from the joint angles: x = L1·cos(θ1) + L2·cos(θ1+θ2) and y = L1·sin(θ1) + L2·sin(θ1+θ2). This is forward kinematics — given the joint angles, find where the tool is. Fundamental in manipulator control and robot simulation. Enter the link lengths (L1, L2) and the joint angles (θ1, θ2) in degrees.
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Cinemática direta (2 GL)
A cinemática direta responde à pergunta: dados os ângulos das juntas de um braço robótico, onde está a ponta da ferramenta? Para um manipulador planar de 2 graus de liberdade (dois elos, duas juntas rotativas), a posição vem da soma vetorial dos elos: x = L1·cos(θ1) + L2·cos(θ1+θ2) e y = L1·sin(θ1) + L2·sin(θ1+θ2). Note que o segundo elo gira em relação ao primeiro, por isso seu ângulo é θ1+θ2 no referencial fixo. A cinemática direta é sempre única e fácil; o problema inverso (achar os ângulos para uma posição desejada) é o difícil — pode ter duas soluções (cotovelo para cima ou para baixo) ou nenhuma. Informe os comprimentos dos elos e os ângulos das juntas.
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Robot Arm Reach
Calculate the maximum reach of a planar two-link robotic arm, R = L₁ + L₂, by adding the lengths of the two links (arm and forearm). The result, in the length unit, is the radius of the work envelope — the farthest distance the tip (end-effector) can reach when the arm is fully extended. It defines the robot's workspace and is the first sizing parameter of manipulators. The minimum reach (dead zone at the center) is |L₁ − L₂|, and the useful area is the annulus between the two radii. Enter the two link lengths.
Servo Torque for Arm
Calculate the static torque a servomotor needs to hold a horizontal arm, T = m·g·L, from the tip mass m, gravity g (9.81 m/s²) and the arm length L. The result, in N·m, is the minimum torque the servo must provide to keep the arm horizontal against the load weight — the most unfavorable position. It is essential in designing robotic arms, grippers and servo-driven mechanisms, sizing the motor with a safety margin over this value. For arms with their own mass, the center of mass is used. Enter the mass and the arm length.
Forward-Start Option
Computes the price of a forward-start call: an option granted now, but whose strike is only set on a future date, usually as a proportion of the price at that moment. It's the structure behind employee option plans and cliquets, where new at-the-money options are issued periodically. Since the strike tracks the future price, the value doesn't depend on the current level in a trivial way. Enter the spot price, the moneyness, the rate, the volatility, the grant date and the expiry.
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