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.
Result
—
Robotic arm reach
A robot's workspace — the region its end effector can actually reach — is the first specification of any manipulator. For a planar two-link arm (the textbook configuration, with an 'upper arm' and a 'forearm' joined by revolute joints), the maximum reach is simply the sum of the link lengths: R = L₁ + L₂, achieved when the arm is fully extended in a straight line. The workspace, however, is not a solid disc — it is a circular annulus (a ring). That is because there is also a minimum reach: the arm cannot serve points very close to its own base, since the forearm has to fold back over the upper arm. That inner radius — the central 'dead zone' — is R_min = |L₁ − L₂|. When both links are the same length (L₁ = L₂) the minimum reach falls to zero and the robot can touch its own centre, which is why equal links are a common design choice. The usable workspace is therefore the annulus between R_min and R_max. Inside it, dexterity still has to be considered: near either limit (arm almost straight or almost folded back) the manipulator loses manoeuvrability and force in certain directions — configurations close to singularities — so the region that is genuinely useful is smaller than the geometric ring. For 3D robots, with more joints and a rotating base, the workspace becomes a volume (typically a portion of a sphere or a toroid), but the principle carries over: the sum of the link lengths gives the reach, and the joint geometry defines the shape of the envelope. Sizing the links to cover the intended task without overshooting — long arms demand strong motors and give up stiffness and precision — is the central trade-off of kinematic design. Enter the lengths of the two links.
Related Tools
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.
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.
Hand-Arm Vibration A(8)
Calculate the normalized hand-arm vibration exposure A(8), A(8) = a_w·√(t ÷ 8), from the resultant acceleration a_w (m/s²) and the exposure time t (hours). The result, in m/s², normalizes the exposure to an 8-hour shift, allowing comparison with the action and tolerance levels. Prolonged exposure to tool vibration (grinders, breakers, chainsaws) causes hand-arm vibration syndrome, with vascular and neurological damage. Enter the resultant acceleration and the exposure time.
Grinding Energy (Bond Work Index)
Calculate the specific comminution (grinding/crushing) energy by Bond's law, W = 10 × Wi × (1/√P₈₀ − 1/√F₈₀), from the ore's Bond work index Wi (kWh/t), and the particle sizes passing 80% of the product (P₈₀) and feed (F₈₀), in micrometers. The result, in kWh per tonne, is the energy needed to reduce the ore from feed to product size. Comminution is mining's largest energy consumer (up to 50% of the plant). The Wi index characterizes the ore's resistance to fragmentation. It is the basis for sizing mills and energy consumption. Enter the Wi, P₈₀ and F₈₀.
Pump Work (Rankine)
Calculate the specific work consumed by the pump of a Rankine cycle, w_pump = v × (P₂ − P₁), multiplying the liquid specific volume (m³/kg, ~0.001 for water) by the pump pressure rise (kPa). The result, in kJ/kg, is the energy spent pressurizing the condensate before the boiler. Because the liquid is nearly incompressible, this work is tiny compared with the turbine's — which is why the Rankine cycle pumps a liquid (not a gas, as a gas Carnot cycle would), sharply reducing the back work. Enter the specific volume and the pump outlet and inlet pressures.
Percent Cold Work
Calculate the percent cold work (area reduction), %CW = (A₀ − A_f) ÷ A₀ × 100%, from the initial cross-section area A₀ and the final area A_f after cold plastic deformation (rolling, drawing, stamping). The result, in %, shows how much the material was deformed below the recrystallization temperature. Cold work strain-hardens the metal: it raises the yield strength and hardness and lowers ductility as dislocations multiply and tangle. It is the parameter used to control properties before an anneal. Enter the initial and final areas.
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.