What is a Parallel Robot? | Kinematics | 4 Important Types

Parallel Robot

Image Credit:  Humanrobo is licensed under CC BY-SA 3.0

The subject of discussion: Parallel Robot and its features

What is a Parallel Robot? | Parallel Robotics

A parallel robot is a mechanical device that supports a single base or end-effector by using multiple computer-controlled serial chains. The most well-known parallel manipulator consists of six linear actuators that sustain a movable foundation for applications like flight simulators. In other names, it is called the Stewart platform, also known as the Gough-Stewart platform, after the engineers who designed and used it first.

Parallel Robot
Parallel Robot: Abstract Rendering; Image Credit: PantoineHexapod0aCC BY-SA 3.0

The actuators are grouped together on both the base and the Stewart platform. There are articulated robots with identical structures for pushing the robot or one or more manipulator arms. The end effector of this arrangement is directly attached to its base by a variety of (usually three or six) distinct and independent linkages operating concurrently instead of a serial manipulator. There is no consequence of geometrical parallelism.

Parallel Robot Design

A Parallel Robot typically has the following design features:

  1. Each chain is usually short and straightforward, allowing it to be rigid against unnecessary movement. Rather than being continuous, errors in one chain’s placement are averaged with the others.
  2. Each actuator must be able to travel within its own range of motion. Their motion is contained within their individual degrees of freedom.
  3. The consequence of the other chains constrains the off-axis stability of a joint in the parallel robot. The ultimate parallel manipulator is rigid in contrast to its components because of this closed-loop stiffness.
  4. A parallel robot’s static representation is often identical to that of a pin-jointed truss: the connections and actuators sense only stress or compression, with no bending or torque, minimizing the consequences of any flexibility to off-axis forces.
  5. The heavy actuators are frequently centrally placed on a single base platform, with the arm’s movement controlled entirely by struts and joints. Because of the decrease in mass around the limb, the component can be built thinner, resulting in lighter actuators and smoother motions. Therefore, the robot’s average moment of inertia is reduced due to the centralization of mass.

Serial and Parallel Robots | Similarities and Differences

The parallel robot can obtain high rigidity even with a small mass of the manipulator concerning the manipulated amount of charge. This gives leverage to the parallel robots over the serial robots because the latter is exposed to an undesirable amount of flexibility or sloppiness in one joint, which causes a similar sloppiness in the arm. The gap in-between the joint and the end-effector can exacerbate this since there is no way to brace one joint’s movement in contrast to other.

Confined workspace is evident in the case of serial manipulators, subject to the design’s geometrical and mechanical constraints, causing collisions between maximal and minimal lengths of the legs. The existence of uniqueness, therefore, constrains the workspace. Singularities are points where the difference in leg lengths is exponentially less than the variation in direction for other movement trajectories.

In comparison, at a single location, a force (such as gravity) acting on the end-effector creates infinitely large restrictions on the legs, possibly resulting in the manipulator explosion. This means that the concurrent manipulators’ workspaces are usually artificially confined to a specific area where no singularity is known.

Non-linear behavior is yet another disadvantage of parallel robots over serial kinematic chain manipulators. The command used to achieve a linear or circular movement of the end-effector is highly dependent on the workspace position and does not change linearly during the exercise.

These characteristics combine to produce manipulators with a broad range of motion. In contrast to serial manipulators, they can be fast-acting because their rigidity limits their action speed rather than brute force.

Stewart Platform and Delta Robot are two commonly used Parallel Robots in the industry.

Stewart Platform

A Stewart platform is a parallel manipulator with 6-prismatic actuaters, typically hydraulic jacks connected in pairs to 3 locations on the base plate and crossed over to 3 points on the top plate. Universal joints are used to connect all 12 points.

Tools mounted on the top plate can travel in all six degrees of freedom available to a free-suspended body: three linear motions (lateral, longitudinal, and vertical), as well as three rotations (pitch, roll, and yaw).

Parallel Robot: Stewart Platform; Public Domain

Linear hydraulic actuators are widely used in industrial applications because of their unique and straightforward inverse kinematics closed-form approach, as well as their high strength and acceleration. these are often utilized for prototyping and economic robotic purposes. Robert Eisele has demonstrated that there is a particular closed-form solution for the inverse-kinematics of rotary actuator.

These system applied in flight simulator, machine tool technologies, crane applications, marine technology, tsunami modelling, air-to-sea rescue operation, mechanical bull, satellite-dish aligning, the Hexapod-Telescope, robots, and others.

Delta Parallel Robot

A delta robot is a parallel robot with three arms that are joined at the base by universal joints. The primary design characteristics is the use of parallelogram in the arms, which preserves the end effector’s positioning and alignments, hence limits the end platform’s movement to pure translation only (no rotational movement is possible).

All of the actuators are placed on the robot’s base, which is positioned above the workspace. Three middle jointed arms stretch from the floor. A small triangular base connects the ends of these cannons. The input ties will move the triangular platform in one of three directions: X, Y, or Z. Linear or rotational actuators may be utilized for actuation.

Parallel Robot: Portrait drawing Delta Robot; Image Credit: Andy DingleySketchy, portrait-drawing delta robotCC BY-SA 3.0

The arms should be made of light plastic material since the actuators are all situated in the base. As a result, the Delta robot’s rotating parts have very little inertia. This enables breakneck speeds and accelerations. The robot’s stiffness is increased by connecting all of its arms to the end-effector, but its working volume is reduced.

There are many different versions of the Delta parallel robot that have been developed over time:

  1. Reymond Clavel produced a version of 4 degrees of freedom:  3-translations and 1-rotational and The 4th leg stretches from the base to the center of the triangular platform, causing the end-effector to rotate around the vertical axis.
  2. Fanuc has designed a Delta robot with 6 degrees of freedom with a serial kinematic with 3 degrees of freedom in rotational motion is to be found on the end effector.
  3. Adept built a Delta robot with four degrees of freedom and a four-parallelogram end effector that is directly connected to its base. It lacks a fourth leg that runs along the center of the end-effector.
  4. Asyril SA built the Pocket Delta robot, which is designed for versatile component feeding systems and other high-speed, high-precision applications.
  5. Delta Direct Drive has 3 degrees of freedom in which the motor is directly attached to the arms. This produces very high accelerations ranging from 30 to 100 g.
  6. Delta Cube was created in a monolithic configuration of flexure-hinges joints by EPFL University Laboratory LSRO. This robot is designed for applications requiring extreme precision.

Delta Parallel Robot finds its applications in 3D printers with a linear delta design, a novel feature. The packaging business, as well as the medical and pharmaceutical industries, also find the Delta parallel robot useful because they benefit from its high speed. It is also used in medicine due to its stiffness. Other uses require electronic part assembly in a clean room with high precision.

3 RRR Planar Parallel Robot

The following image corresponds to a 3 RRR Parallel Robot:

Image Credit: ResearchGate

5 Bar Parallel Robot

The following image corresponds to a 5 Bar Parallel Robot:

Parallel Robot Kinematics

Kinematics of the Moving Platform

The Stewart Platform is the most popular choice among parallel robot manipulators for industrial use. Because of its implementation to 6 degree-of-freedom machine tool architectures, the Stewart Platform mechanism has gotten a lot of coverage. Unlike traditional machine tool architectures, the mechanism’s excellent mechanical features, such as higher rigidity, strength-to-weight ratio, and more excellent maneuverability, have inspired such an application. Let us try to understand the kinematics of a parallel robot through the Stewart Platform.

The Stewart Platform is made up of a payload platform and six linear actuators or struts. The struts’ other ends are fixed to the foundation. A 3o freedom joint and a 2o freedom joint, or 2-3 degree freedom joint, join one of the struts to the foundation and the floor. The platform has six degrees of freedom, with three translational and three rotational degrees of freedom, owing to the six actuators’ linear extension and retraction.

Parallel Robot: Coordinate Frames; Image Credit: ResearchGate

The inverse Jacobian matrix and its time derivatives are determined using closed-form expressions in the kinematic analysis. The effects of various strut end joint arrangements are studied through precise modeling on the mechanism’s kinematics and dynamics.

To reflect rigid body kinematics and dynamics, Euler angles are commonly used. We’ll use a series of Euler angles (φ, θ, ψ) to compute the direction of a rigid-body after the consequent rotational-chains:

  1. The rotation of φ arround the Z’-axis of the moving co-ordinate system.
  2. The rotation of θ arround the x’-axis of the moving co-ordinate system.
  3. The rotation of ψ arround the z”-axis of the moving co-ordinate system.
Parallel Robot: Euler Angles; Image Credit: ResearchGate

W is the world coordinate frame and P is defined as position vector x=(X, Y, Z)T, which is attached to the moving platform at reference point po.

q is a generalized coordinate vector with six variables to define the position and orientation of the Stewart platform, given as:

q=(X, Y, Z, \phi , \theta , \psi )^{T}

And the joint space coordinate is defined as

l=(l_{1},l_{2},l_{3},l_{4},l_{5},l_{6})^{T}

wRp is the orientation of frame P with respect to W by a rotation matrix, wRp = (r1,r2,r3), where r1,r2,r3 are 3×1 unit vectors with respect to P and W. The frame X’-Y’-Z’ is a non rotational coordinate system that interprets with the rigid-body. The frame x-y-z is a body coordinate system that rotates as well as translates  in respect to rigid-body. The latter’s mapping on the former coordinate system is developed through a 3×3 rotation matrix involving wRp and φ, θ, ψ.

^{w}R_{p} = \begin{bmatrix} c\psi c\phi -c\theta s\phi s\psi & -s\psi c\phi -c\theta s\phi c\psi & s\theta s\phi \\ c\psi c\phi +c\theta s\phi s\psi & -s\psi c\phi +c\theta s\phi c\psi & -s\theta c\phi\\ s\psi s\theta & c\psi s\theta & c\theta \end{bmatrix}

Where c and s denote cosine and sine, respectively.

The angular velocity is expressed as ω= (ωx, ωy, ωz)T and α= (αx, αy, αz)T is the angular acceleration concerning W. Therefore, upon resolving the first and second time derivatives of the Euler angles gives us,

\omega = \begin{bmatrix} \omega _{x}\\ \omega _{y}\\ \omega _{z} \end{bmatrix} = \begin{bmatrix} 0 & c\phi & s\phi c\theta \\ 0 & s\phi & -c\phi s\theta \\ 1 & 0 & c\theta \end{bmatrix}.\begin{bmatrix} \dot{\phi} \\ \dot{\theta }\\ \dot{\psi} \end{bmatrix}

And

\alpha = \begin{bmatrix} \alpha _{x}\\ \alpha _{y}\\ \alpha _{z} \end{bmatrix} = \begin{bmatrix} 0 & c\phi & s\phi s\theta \\ 0 & s\phi & -c\phi s\theta \\ 1 & 0 & c\theta \end{bmatrix}.\begin{bmatrix} \ddot{\phi}\\ \ddot{\theta}\\ \ddot{\psi} \end{bmatrix} + \begin{bmatrix} 0 & \dot{\phi}s\phi & \dot{\phi}c\phi s\theta + \dot{\theta}s\phi c\theta\\ 0 & \dot{\phi}c\phi & \dot{\phi}s\phi s\theta - \dot{\theta}c\phi c\theta\\ 0 & 0 & -\dot{\theta}s\theta \end{bmatrix}. \begin{bmatrix} \dot{\phi} \\ \dot{\theta }\\ \dot{\psi} \end{bmatrix}

Parallel Robot Inverse Kinematics

The Stewart Platform’s inverse kinematic problem involves calculating the displacements of the six connections and their time derivatives for a given Cartesian pose of the moving platform, which has variables as three positional displacements and three Euler angular displacements and their time derivatives.

ai is the ith attachment point in P with respect to W, given as:

a_{i} = x + ^{w}R_{p} . ^{p}a_{i}

Then vector Li or the ith link is obtained as:

L_{i} = a_{i} - b_{i}

Where bi is the base attachment point.

Thus the link length li of the ith link is determined as:

l_{i} = \sqrt{L_{i}. L_{i}}

The unit vector along the axis of the prismatic joint of link i is calculated as follows

n_{i} = L_{i} / l_{i}

What are Parallel Robots used for? | Parallel Robot Applications

Parallel Robot finds many applications in the industry, with the conventional Parallel robot manipulator having uses popularly in these devices:

  1. Simulators of flight
  2. Workplace simulations of automobiles
  3. optical fibre orientation / photonics
  4. Micromanipulators mounted on end effectors of larger but slower serial manipulators.
  5. High speed/high-precision milling machines in high speed
  6. High-accuracy placement with minimal workspace, such as in PCB assembly

Parallel robots have a minimal workspace as they can’t reach through barriers. The calculations used to carry out the desired manipulation (forward kinematics) are typically more complex. They can result in several solutions, thereby making this problem beyond this article’s scope.

Serial vs Parallel Robots | Are Parallel Robots more accurate than Serial Robots?

  • Parallel robots are said to be inherently more reliable than serial robots because their errors are averaged rather than applied cumulatively.
  • Input errors are less sensitive in parallel robots than in serial robots. However, this analogy’s scope is too narrow to draw any broad conclusions.

Furthermore, making a meaningful distinction between other pairs of serial and parallel robots is almost impossible. As a result, there is no clear solution to the dominance argument.

About Esha Chakraborty

I have a background in Aerospace Engineering, currently working towards the application of Robotics in the Defense and the Space Science Industry. I am a continuous learner and my passion for creative arts keeps me inclined towards designing novel engineering concepts.
With robots substituting almost all human actions in the future, I like to bring to my readers the foundational aspects of the subject in an easy yet informative manner. I also like to keep updated with the advancements in the aerospace industry simultaneously.

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