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A novel three-legged 6-DOF parallel robot with simple kinematics

Publication: Transactions of the Canadian Society for Mechanical Engineering
4 March 2020


This paper presents a novel three-legged six degrees of freedom (6-DOF) parallel robot with simple kinematics. The main idea behind this novel architecture is that each of the three identical legs is controlled by two prismatic actuators with parallel directions. As a result, it is possible to control simultaneously or separately the position and the orientation of a leg. The reduced number of legs leads to a simple mechanical design with reduced risk for mechanical interferences.


Cet article présente un nouveau robot parallèle à six degrés de liberté (6-ddl), avec trois jambes et un modèle cinématique simple. L’idée principale de cette nouvelle architecture est que chacune des trois jambes identiques est contrôlée par deux actionneurs linéaires dont les directions sont parallèles. De ce fait, il est possible de contrôler simultanément ou séparément la position et l’orientation d’une jambe. Le nombre réduit de jambes permet une conception mécanique simple, avec un faible risque d’interférences mécaniques. [Traduit par la Rédaction]

1. Introduction

Many robots with parallel kinematics have been invented, but none of them is as simple as the Cartesian parallel robot (Kong and Gosselin 2002), also known as Tripteron (Gosselin et al. 2004). The kinematic model of the latter is simply
where ρ1, ρ2, and ρ3 are the active-joint variables; and x, y, and z are the position coordinates of the mobile platform.
Tripteron was a starting point for developing a whole new family of parallel robots. The concept was first extended to a four-legged robot with four degrees of freedom (4 DOF), called Quadrupteron (Richard et al. 2006), but its direct kinematic problem requires the solution of a quadratic equation. Then, the next member of the Cartesian parallel robot family was invented, the Pentapteron (Kong and Gosselin 2005; Gosselin et al. 2007). The latter has 5 DOF, but its direct kinematic problem allows at least 208 real solutions (Masouleh et al. 2010). Later, a 6-DOF member was presented (Seward and Bonev 2014), the Hexapteron, which surprisingly has simple direct and inverse kinematics.
In this article, we propose a new 6-DOF Cartesian robot with only three legs (Fig. 1). Whereas the design of Hexapteron’s legs is intricate, requiring complex five-axis machining to avoid mechanical collisions, our novel architecture allows the design to be kept simple and requires only three-axis machining.
Fig. 1.
Fig. 1. Schematic of the novel 6-DOF parallel robot. [Colour online.]
First, we study the kinematic equations of the new robot, which are still trivial as in the case of the Hexapteron. Second, we analyze the workspace of the robot. Finally, we show how to transform our architecture easily to 5-DOF and 4-DOF alternatives or other 6-DOF designs while keeping the simplicity of the kinematic equations.

2. Kinematic modelling of the robot

The schematic of the novel architecture is shown in Fig. 1. Legs 1, 2, and 3 are each attached to the base by a set of two active prismatic joints with directions parallel to the x, y, and z axis of the base (Cartesian) reference frame, respectively. Each of the two carriage blocks of the first, second, and third pair of linear guides are connected to secondary blocks through passive revolute joints with axes parallel to axes z, x, and y, respectively. The outer block of each pair of secondary blocks (i.e., the secondary block of legs 2, 4, and 6) is connected to the inner block through a passive prismatic joint, the direction of which is normal to the axes of the two passive revolute joints just described. This type of actuation has already been used in Yu et al. (2006). Then, the proximal link of each leg is connected to the inner secondary block and to the corresponding distal link via revolute joints, the axes of which are parallel to each other, and normal to the axes of the passive revolute joints previously described. Finally, legs 1, 2, and 3 are each connected to the mobile platform via a passive universal joint, the first axis of which is parallel to the axes of the preceding two revolute joints, and the second axis is along the y, z, and x axis of the mobile (Cartesian) reference frame, respectively.
To illustrate the behavior of a leg, we can distinguish two separate cases. The first case is when the difference of two adjacent actuator (active-joint) variables is constant. As a result, the plane of the leg (see Fig. 2) will only translate along the direction of the corresponding prismatic joint. The second case is when the difference of the two active-joint variables varies and both secondary blocks in a leg move with respect to one another. Then the plane of the leg will translate and simultaneously rotate about the axis of the passive revolute joint in the inner secondary block.
Fig. 2.
Fig. 2. Schematic of the proposed parallel robot showing the leg planes πi. [Colour online.]
Let π1, π2, and π3 be the leg planes (Fig. 2), each normal to the axes of the three intermediate revolute joints and passing through the axes of the other three revolute joints of a leg. Thus, π1 contains axis y′ of the mobile reference frame and is parallel to axis z of the base reference frame, π2 contains axis z′ and is parallel to axis x, and π3 contains axis x′ and is parallel to axis y.

2.1. Inverse kinematics

For clarity, refer to Fig. 3, which shows the projections of the leg planes onto the base reference frame planes. For the inverse kinematic problem, we have to find the length of segments DiPi, i.e., the active-joint variables ρi (in this paper i = 1, 2, …, 6). Let γ1, γ2, and γ3 be the angles between planes π1, π2, and π3 and the planes Oyz, Oxz, and Oxy, respectively (see Figs. 2 and 3). Thus, we have
where r1,1, r1,2, etc., are the elements of the rotation matrix R representing the orientation of the mobile reference frame with respect to the base reference frame.
Fig. 3.
Fig. 3. Projections of the auxiliary planes.
Let x, y, and z be the coordinates of point C with respect to the base reference frame. Finally, let di be the distance between point Di and the origin of the base reference frame, and let d1 = d3 = d5 and d2 = d4 = d6.
Thus, it can be easily seen that the active-joint variables are calculated as follows:

2.2. Direct kinematics

To solve the direct kinematics, we first need to find the angles γ1, γ2, and γ3 from the following equations:
Then, we can find the position of the mobile platform by solving the system of linear eqs. 5, 7, and 9:
where t1 = tan γ1, t2 = tan γ2, and t3 = tan γ3. Obviously, there is always a single solution for the position of the mobile platform, except when t1t2t3 = 1. As we will see in section 2.3, the latter corresponds to a singularity.
Now, it is relatively easy to see that the rotation matrix R has the following form:
where α1 is the angle between the mobile y′ axis and the base z axis, α2 is the angle between the mobile z′ axis and the base x axis, and α3 is the angle between the mobile x′ axis and the base y axis. For a practical design, these angles are close to 90° and could never be 0° or 180°, i.e., their sines are always positive. Multiplying each two columns of the rotation matrix and dividing by the corresponding sin αj (j = 1, 2, 3), yields the following three equations (note that sin αj ≠ 0, as αj can never be 0° or 180°):
Finally, we can solve the above three equations for the cotangents and obtain
As we already mentioned, because the absolute values of the angles γ1, γ2, and γ3 are <45°, the denominator in the above expressions is always non-zero. From these three equations, we can find α1, α2, and α3, and therefore find the rotation matrix R, which concludes the direct kinematic problem.

2.3. Velocity kinematics and singularities

To find the velocity equations, we simply need to differentiate eqs. 510 with respect to time:
The remarkable fact about this new parallel robot is that , , and , where ω = [ωx, ωy, ωz]T is the vector of angular velocity of the mobile platform. Therefore, the following matrix velocity equation can be written:
where the 6 × 6 matrix is the inverse of the Jacobian matrix, denoted as J–1. Finally, to find all singularities, we calculate the determinant of that matrix:
Obviously, for a practical mechanism, the above expression can be zeroed only if 1 – t1t2t3 = 0. Geometrically, this condition holds true when the three planes π1, π2, and π3 intersect at a common line (note that it is impossible for any two planes to coincide). Fortunately, for a typical mechanical design, the absolute values of angles γ1, γ2, and γ3 are <45°, so our mechanism is never at a singularity.

3. Static analyses

Unlike most parallel mechanisms, the Jacobian matrix of our parallel mechanism can be calculated analytically and has a fairly simple form:
where t123 = t1t2t3 and d12 = d1d2.
The relationship between the joint forces, τ, and the wrench, F, is
From here, we can see, for example, that a torque applied to the mobile platform, about axis x, is resisted only by actuators 1 and 2. Furthermore, as the value of d12 is relatively small, high actuator forces are necessary to resist small torques. In contrast, a force applied to the mobile platform is resisted by all actuators, though mostly by two actuators in parallel. Therefore, this parallel mechanism has relatively low resistance to external forces applied to its mobile platform, and even lower resistance to external torques, especially when compared to the typical telescoping-legs hexapod.

4. Workspace

For the studied architecture, each leg is able to perform two types of motions, a linear motion and a rotation about an axis perpendicular to the direction of its prismatic joint. When the mobile platform has zero orientation, the robot behaves exactly as the Tripteron. The only difference is that even for pure translations along x, y, and z axes, all six motors are controlled in pairs. In this mode, all legs move only linearly along the rails, without simultaneous rotation. Thus, for the pure translation mode, the workspace is a cube with a side equal to the stroke of the actuators, or slightly less (when the platform is rotated).
For 6-DOF applications such as pick-and-place or assembly, it is difficult to represent the orientation workspace of the robot, as it varies significantly from point to point. Referring to eq. 17 and Fig. 2, we can obtain some rough indications about the ability to rotate about some axes, as one of the main mechanical limits is the limit on angles γ1, γ2, and γ3. However, as we will be interested in orienting a specific tool reference frame, not the mobile Cxyz′ reference frame, it is very difficult to discuss the orientation workspace of a general case.
For machining or additive manufacturing applications, however, we are only interested in two orientation DOF (the rotation about the axisymmetric tool is redundant). For such five-axis applications, the tool must be placed as in Fig. 4, to gain advantage of the inherent symmetry of the parallel robot. A video illustrating the proposed robot design is available at
Fig. 4.
Fig. 4. Proposed mechanical design of the novel parallel robot for machining applications. [Colour online.]
For representing the workspace of the specific implementation shown in Fig. 4, we will therefore need a new set of base and mobile reference frames. These are defined in Fig. 5. Finally, we will use the tilt and torsion angles (Bonev and Ryu 2001) to represent the orientation of the mobile platform. An orientation is defined with these angles as follows: From its zero orientation, rotate the mobile reference xyz′ about an axis in the xy′ plane, making an angle with axis x′, at θ (the tilt angle), and then rotate about the z′ axis at σ (the torsion angle). As shown in Bonev and Ryu (2001), for symmetric parallel robots such as the one shown in Fig. 4, maximum tilt is obtained at torsion angles that are very close to zero. Therefore, we can greatly simplify the workspace analysis and the operation of the robot by keeping the torsion angle equal to zero. Finally, for a given position of the tooltip, although the mobile platform can tilt in some directions more than in others (i.e., for some angles ), we are interested in the maximum tilt angle that can be achieved in any direction (i.e., for any angle ).
Fig. 5.
Fig. 5. New base and mobile reference frames for the robot shown in Fig. 4.
Unfortunately, it is impossible to devise a geometric algorithm for computing the maximum tilt angle for a given tooltip position. We therefore use a simple discretization algorithm, and for each tooltip position of a Cartesian grid, we vary from 0° to 360°, and for each , we increase θ from 0° until a mechanical interference is reached. The mechanical limits that we test for are angles γ1, γ2, and γ3 (40°) and the actuator strokes (500 mm).
Figure 6 presents the results, by displaying 12 different “horizontal” sections of the position workspace of the proposed robot. For each section, a color map is used to show the values for the maximum tilt angles.
Fig. 6.
Fig. 6. Maximum tilt angle (in degrees) at different sections of the position workspace of the proposed robot. [Colour online.]

5. Alternative designs

Although the studied architecture has 6 DOF, architectures with fewer DOF might also be achieved using the same main idea. Conventional spatial parallel kinematic machine tools (PKMs) have 5 DOF: three translational DOF and two rotational. The axis for the third rotational movement coincides with the rotational movement of the cutting tool. So for reducing the mechanical complexity and cost, most PKM are built without the ability for third rotational DOF.
In Fig. 7a, we propose a 5-DOF version that can be used for machining or additive manufacturing. The 5-DOF version is achieved by removing motor 4 and blocking the rotation of leg 1. Thus, leg 1 is only able to translate along its rail. We can also transform our 6-DOF robot to a 4-DOF robot, as shown in Fig. 7b. This kind of architecture is useful for pick and place operations.
Fig. 7.
Fig. 7. Reduced-DOF alternative designs: (a) 5-DOF version and (b) 4-DOF version. [Colour online.]
Finally, in this paper, we studied a design in which both the directions of the prismatic actuators and the axes of the platform revolute joints are orthogonal. Here, we propose two other geometries. In the first one, shown in Fig. 8a, the angle between the directions of each pair of prismatic joints and the horizontal plane is 15°. Note that this angle cannot be 0° or else the robot will always be in a singularity. However, the axes of the platform revolute joints are coplanar. This design will be better suited for machining, as torques about the tool will be resisted by all actuators, in a uniform fashion.
Fig. 8.
Fig. 8. Other designs with different orientations of the base prismatic joints and platform revolute joints. [Colour online.]
In the second geometry, shown in Fig. 8b, the directions of the prismatic joints are vertical. In this design, the axes of the platform revolute joints are not coplanar or else the robot would be always at a singularity. The angle between each pair of platform revolute joints is 110°. Unfortunately, such a mechanism would have low resistance to torques about its tool.
The equations for the inverse and direct kinematic problems for the proposed two new designs are like those shown in section 2. However, the singularities of the new designs are substantially different. It is therefore crucial to perform an optimal design before selecting a specific design.

6. Conclusions

A novel three legged 6-DOF parallel robot was proposed. The mechanism has very simple inverse and direct kinematic problems, with unique solutions. Both the Jacobian and the inverse Jacobian matrices have relatively simple form. The singularities of this mechanism are also easy to define, and impossible to attain for a practical design. Some subsets of the workspace of the mechanism were also computed and represented. It was found that the mechanism has a relatively large workspace for machining applications that do not require large tilt angles. That said, a static analysis of the mechanism showed that the proposed mechanism has relatively low resistance to external forces applied to its end-effector, and even lower resistance to external torques.
Finally, 5-DOF and 4-DOF variants, as well as other 6-DOF designs with different angular dimensions were proposed. One of the 6-DOF designs is probably well-suited for machining operations and deserves further attention.


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Information & Authors


Published In

cover image Transactions of the Canadian Society for Mechanical Engineering
Transactions of the Canadian Society for Mechanical Engineering
Volume 44Number 4December 2020
Pages: 558 - 565


Received: 26 July 2019
Accepted: 9 January 2020
Accepted manuscript online: 4 March 2020
Version of record online: 4 March 2020


This paper is part of a Special Issue with the best papers presented during the 2019 CCToMM Symposium on Mechanisms, Machines, and Mechatronics.

Key Words

  1. Cartesian parallel robot
  2. direct kinematics
  3. workspace
  4. PKM


  1. robot parallèle cartésien
  2. cinématique directe
  3. espace de travail
  4. PKM



Edelvays Cherchelanov
École de technologie supérieure (ÉTS), 1100 rue Notre-Dame Ouest, Montreal, QC H3C 1K3, Canada
Ilian A. Bonev [email protected]
École de technologie supérieure (ÉTS), 1100 rue Notre-Dame Ouest, Montreal, QC H3C 1K3, Canada


Copyright remains with the author(s) or their institution(s). This work is licensed under a Creative Commons Attribution 4.0 International License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.

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