Parallel_manipulator - Wikiparency

# Parallel manipulator A **parallel manipulator** is a mechanism that positions a platform, or end-effector, on several independent kinematic chains connected in parallel between the platform and a fixed base, rather than the single open chain of a [[Serial_manipulator|serial manipulator]].[^merlet2006] Sharing the load across several legs makes the mechanism stiff, accurate and fast to move; the trade is a smaller, more awkwardly shaped reachable region, and a mathematics problem that runs backwards from a serial arm's.[^merlet2006] The best-known parallel manipulator carries its platform on six adjustable-length legs, giving it all six spatial [[Degrees_of_freedom_(mechanics)|degrees of freedom]] a rigid body can have — three of position and three of orientation.[^dof199] Widely known as the [[Stewart_platform|Stewart platform]], or the Gough–Stewart platform, it takes its names from Eric Gough, a tyre engineer who built an early six-legged rig to load-test tyres under combined forces, and from D. Stewart, whose 1965 paper proposed the same six-actuator arrangement independently, as the moving base of an aircraft simulator.[^stewart1965] The name also covers the wider family: three-legged and planar parallel mechanisms move a platform with fewer freedoms, and the [[Delta_robot|delta robot]], built from parallelogram legs, trades rotation away entirely to move a small platform in pure translation at high speed. That backwards mathematics is exact for the six-leg platform. Each leg's length follows directly from the platform's commanded position and orientation, a short, closed-form calculation; the reverse problem — recovering the platform's pose from six measured leg lengths — generally has no closed form and can have more than one geometrically valid answer.[^merlet2006] A serial manipulator has the opposite problem, easy forward kinematics and hard [[Inverse_kinematics|inverse kinematics]], so the two families specialise in solving each other's difficult half.[^merlet2006] A platform command is reachable only if every leg can still reach its two joints without running past its shortest or longest length, so the mechanism's reach is bounded by its legs' stroke, not by a simple maximum radius. The framework microsim *Parallel manipulator: six legs, one platform* builds exactly this six-leg platform in three dimensions, drives it to a commanded pose, and colours each leg by how close it sits to that stroke limit — the moment the sim, and the real mechanism, must stop. ## History Two engineers are usually credited with the parallel manipulator's modern form, arriving at it from opposite directions. Eric Gough built an early hexapod rig with six adjustable legs to load-test aircraft tyres under combined vertical, lateral and torsional loads, working from a practical testing problem rather than from theory.[^merlet2006] D. Stewart's 1965 paper, "A Platform with Six Degrees of Freedom," proposed the same six-leg arrangement independently: six actuators under a platform, driven together, could reproduce an aircraft's motion in all six degrees of freedom at once, in a far smaller footprint than the gimballed motion platforms flight simulators had used before.[^stewart1965] Later writers folded the two names together as the [[Stewart_platform|Gough–Stewart platform]], and the device became the moving base of nearly every modern flight and vehicle simulator.[^merlet2006] The broader term *parallel manipulator*, covering any mechanism built from several legs connected in parallel rather than specifically six, came into use as engineers designed machine tools, medical robots and other mechanisms on the same closed-loop principle, some carrying fewer than six legs and fewer than six degrees of freedom.[^merlet2006] ## Design A parallel manipulator is built from three parts: a fixed base, a moving platform, and the legs that connect them. Each leg is usually a chain of a universal joint at the base, letting it swing in two directions; a linear actuator that changes the leg's length; and a spherical joint at the platform, letting it swing freely — a universal–prismatic–spherical, or UPS, leg, whose six copies, in the common six-leg design, are together enough to fix the platform's full six-degree-of-freedom pose.[^merlet2006] ### Configuration The six base joints and six platform joints are rarely spaced evenly around their circles; a common arrangement groups them into close pairs, three around the base and three around the platform, offset from each other so that alternate legs cross rather than run parallel — the "6-6" layout.[^simspec] The framework sim uses exactly this layout: three base-joint pairs 20 degrees apart, centred 120 degrees apart around the base, and three platform-joint pairs 100 degrees apart, centred the same way around the platform.[^simspec] Other real designs space the joints differently, including a "6-3" layout with three widely separated base joints and three closely paired platform joints; the joint spacing changes the platform's stiffness and its resistance to particular loads without changing the underlying six-leg mathematics.[^merlet2006] Reducing the leg count below six trades away degrees of freedom for a simpler mechanism: the [[Delta_robot|delta robot]]'s three parallelogram legs, for instance, remove all three rotational freedoms and leave only the three translations a pick-and-place task needs. ### Kinematics For the six-leg platform, leg *i* joins a base joint b_i to a platform joint p_i, and its length is simply the distance between them once the platform has been moved to a pose x (surge, sway, heave) and turned through a rotation R built from roll, pitch and yaw:[^simspec] `l_i = | x + R(roll, pitch, yaw) p_i - b_i |, i = 1..6` Computing all six lengths from a commanded pose — the inverse kinematics — is nothing more than evaluating that equation six times. Computing the pose from six measured lengths — the forward kinematics — inverts the same equation and is far harder: it generally has no closed-form solution, can admit as many as forty geometrically valid poses for a general Gough–Stewart geometry, and in practice is solved numerically or by adding extra sensors that measure the pose more directly.[^merlet2006] Every leg can extend only over its actuator's stroke, a fixed range around its length at the platform's home pose; a leg that reaches either end of that range while chasing a command stops the platform there, short of the target, which is what sets the outer edge of the platform's reachable region.[^simspec] *Try: a Stewart platform's pose has six degrees of freedom — surge x, sway y, heave z, roll, pitch and yaw; this build holds sway at zero and gives sliders for the other five plus the stroke window. Drag heave z and pitch together and watch all six bars shift together; then push yaw toward its 60-degree limit and watch alternating legs stretch long and pull short as the platform twists, one bar crossing into red, the count of legs at their limit climbing, and "in workspace" flip from yes to no as the platform stops short of the command.* ## Applications Stewart's own proposal was a flight simulator, and that remains the parallel manipulator's most familiar job: a six-leg platform under a cockpit mock-up reproduces an aircraft's roll, pitch, yaw and translations from data driven by a flight model, at a cost and footprint far below the gimballed rigs it replaced.[^stewart1965] The same six-actuator base now moves driving simulators, amusement rides and other motion platforms built to reproduce a vehicle's felt motion rather than to move it any real distance. Away from simulation, the same stiffness that makes a parallel manipulator awkward to reach far with makes it attractive for cutting metal: parallel-kinematic machine tools carry a spindle on a Stewart-platform-like base instead of the stacked slides of a conventional milling machine, trading a smaller work envelope for a lighter, stiffer structure that can move and cut faster.[^merlet2006] Smaller parallel mechanisms, including delta-robot designs, position surgical tools and other precision instruments where a light, stiff, fast-moving platform matters more than a large reach. The word "hexapod" names two different machines that share only a leg count. A Stewart platform is sometimes called a hexapod for its six actuated legs, none of which ever touch a floor; a [[Hexapod_(robotics)|hexapod robot]] is a six-legged walking machine whose legs lift and place in a walking gait, the subject of Wikitube's [[Legged_robot|legged-robot]] section on locomotion and gaits. A walking hexapod's legs that are on the ground at any instant form a closed loop with the body, in the same structural sense that a Stewart platform's six legs always do; Wikitube's section on legged locomotion analyses exactly that loop's balance with [[Zero_moment_point|zero-moment-point]] methods. ## See also - [[Stewart_platform]] - [[Delta_robot]] - [[Hexapod_(robotics)]] - [[Serial_manipulator]] - [[Inverse_kinematics]] - [[Degrees_of_freedom_(mechanics)]] - [[Zero_moment_point]] - [[Legged_robot]] ## References [^merlet2006]: Merlet, Jean-Pierre. *Parallel Robots*, 2nd ed. (Springer, 2006), chs. 2–5: the general inverse kinematics of a Gough–Stewart platform, its 6-6 and 6-3 joint layouts, forward-kinematics solution counts, and the wider family of parallel mechanisms built on the same closed-loop principle, including the delta robot and parallel-kinematic machine tools. [^stewart1965]: Stewart, D. "A Platform with Six Degrees of Freedom." *Proceedings of the Institution of Mechanical Engineers* 180, pt. 1, no. 15 (1965): 371–386: the six-actuator platform proposed as the moving base of an aircraft simulator. [^dof199]: Jensen, David. *Introduction to Mechanical Design and Manufacturing* (2024). Portal Book 109, p. 199: a free rigid body in three-dimensional space has six degrees of freedom, three of translation and three of rotation. [^simspec]: Robotics portal sim spec `specs/sims/Parallel_manipulator.json`, framework module `robot.kin`: `stewartAngles` gives the 6-6 joint layout (base pairs 20 degrees apart, centred at 0, 120 and 240 degrees; platform pairs 100 degrees apart, centred the same way), `stewart` returns the six leg lengths l_i = |x + R(roll, pitch, yaw) p_i - b_i| for a commanded pose, and `util.bisect` finds where the straight path from home to the command first meets a leg's stroke limit. Geometry: base radius 1.0 m, platform radius 0.7 m. The stroke window is the dialled stroke fraction of the home pose's own leg length, split evenly above and below it. Nothing in the sim is ILLUSTRATIVE; the base and platform radii and the 6-6 half-angles are one representative layout and do not change the kinematics. **Microsim — three.js (Wikitube framework):** *Parallel manipulator: six legs, one platform*  *Built from `MICROSIM_GUIDE/specs/sims/Parallel_manipulator.json`; part of the [[PORTAL_Robotics|Robotics portal]] spine (section ROB5, "Parallel manipulators"). Pending the robotics set's production deploy — see the portal's work list.* ## Wikipedia : Wikitube **Strict pair:** [Wikipedia](https://en.wikipedia.org/wiki/Parallel_manipulator) : [Wikitube](https://en.wikitube.io/wiki/Parallel_manipulator) - skeleton pinned to revision 1360716208 (2026-09-23). *Editorial note: live Wikipedia access was unavailable to this authoring session — organization network policy blocked `*.wikipedia.org` on both the Mac's and the container's network lanes (curl returned HTTP 403 from the proxy on every host tried, including `en.wikipedia.org`, `api.wikimedia.org` and `en.m.wikipedia.org`), and the Mac's own `parity_check.py --outline "Parallel manipulator"` hung with no output on its dedicated Wikipedia network lane; the WebFetch tool's permission gate also had no user available to approve it in this headless run. The section skeleton above (History, Design, Applications) was reconstructed from established robotics-literature knowledge of this article's subject rather than read from the pinned revision directly. Re-run `parity_check.py "Parallel manipulator"` against a working network before promoting this page past `status: draft`.*