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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Forward kinematics (2 DOF)
Forward kinematics answers the question: given the joint angles of a robot arm, where is the tool tip? For a planar manipulator with two degrees of freedom (two links, two revolute joints), the position follows from the vector sum of the links: x = L1·cos(θ1) + L2·cos(θ1+θ2) and y = L1·sin(θ1) + L2·sin(θ1+θ2). Note that the second link rotates relative to the first, which is why its angle in the fixed frame is θ1+θ2. Forward kinematics always has a unique answer and is easy to compute; the inverse problem (finding the joint angles that reach a wanted position) is the hard one — it may have two solutions (elbow up or elbow down) or no solution at all. Enter the link lengths and the joint angles.
Related Tools
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.
FRA Settlement (Forward Rate Agreement)
Computes the settlement amount of an FRA (Forward Rate Agreement), the contract that locks in today an interest rate for a future period. At fixing, the contracted rate is compared with the market reference rate, and the difference is paid at the start of the period — which is why it's discounted: N·(L − R)·(d/B)/(1 + L·d/B). When the market rate exceeds the contracted one, the party who locked in gains. It's used to hedge loans and deposits against rate moves. Enter the notional, the contracted rate, the reference rate, the days in the period and the day-count basis.
Trapezoidal Profile Time
Compute the total time of a motion with a trapezoidal velocity profile, t = d/Vmax + Vmax/a, adding the cruise-velocity time to the acceleration and deceleration phases. It is the most common motion profile in motors and robots: accelerate to maximum speed, hold constant and decelerate. Enter the distance, the maximum velocity and the acceleration (assuming equal acceleration and deceleration).
Servo Angle from PWM
Convert a servo motor's PWM pulse width into the corresponding angle, angle = (pulse − 1000)/1000 · 180°, in the hobby standard where 1000 µs ≈ 0°, 1500 µs ≈ 90° (center) and 2000 µs ≈ 180°. Hobby and robotics servos are commanded by this pulse width, typically at 50 Hz. Knowing the relation helps to calibrate and program movements. Enter the pulse width in microseconds.
The results provided by this tool are for general informational and educational purposes only and do not constitute professional, financial, medical, legal, tax or accounting advice. Always confirm important decisions with a qualified professional and official sources.