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

## Abstract

## Résumé

## 1. Introduction

*ρ*

_{1},

*ρ*

_{2}, and

*ρ*

_{3}are the active-joint variables; and

*x*,

*y*, and

*z*are the position coordinates of the mobile platform.

## 2. Kinematic modelling of the robot

*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.

*π*

_{1},

*π*

_{2}, and

*π*

_{3}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,

*π*

_{1}contains axis

*y*′ of the mobile reference frame and is parallel to axis

*z*of the base reference frame,

*π*

_{2}contains axis

*z*′ and is parallel to axis

*x*, and

*π*

_{3}contains axis

*x*′ and is parallel to axis

*y*.

### 2.1. Inverse kinematics

*D*, i.e., the active-joint variables

_{i}P_{i}*ρ*(in this paper

_{i}*i*= 1, 2, …, 6). Let

*γ*

_{1},

*γ*

_{2}, and

*γ*

_{3}be the angles between planes

*π*

_{1},

*π*

_{2}, and

*π*

_{3}and the planes

*O*–

*yz*,

*O*–

*xz*, and

*O*–

*xy*, respectively (see Figs. 2 and 3). Thus, we have

*r*

_{1,1},

*r*

_{1,2}, etc., are the elements of the rotation matrix

**R**representing the orientation of the mobile reference frame with respect to the base reference frame.

*x*,

*y*, and

*z*be the coordinates of point

*C*with respect to the base reference frame. Finally, let

*d*be the distance between point

_{i}*D*and the origin of the base reference frame, and let

_{i}*d*

_{1}=

*d*

_{3}=

*d*

_{5}and

*d*

_{2}=

*d*

_{4}=

*d*

_{6}.

### 2.2. Direct kinematics

*γ*

_{1},

*γ*

_{2}, and

*γ*

_{3}from the following equations:

*t*

_{1}= tan

*γ*

_{1},

*t*

_{2}= tan

*γ*

_{2}, and

*t*

_{3}= tan

*γ*

_{3}. Obviously, there is always a single solution for the position of the mobile platform, except when

*t*

_{1}

*t*

_{2}

*t*

_{3}= 1. As we will see in section 2.3, the latter corresponds to a singularity.

**R**has the following form:

*α*

_{1}is the angle between the mobile

*y*′ axis and the base

*z*axis,

*α*

_{2}is the angle between the mobile

*z*′ axis and the base

*x*axis, and

*α*

_{3}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

*α*(

_{j}*j*= 1, 2, 3), yields the following three equations (note that sin

*α*≠ 0, as

_{j}*α*can never be 0° or 180°):

_{j}*γ*

_{1},

*γ*

_{2}, and

*γ*

_{3}are <45°, the denominator in the above expressions is always non-zero. From these three equations, we can find

*α*

_{1},

*α*

_{2}, and

*α*

_{3}, and therefore find the rotation matrix

**R**, which concludes the direct kinematic problem.

### 2.3. Velocity kinematics and singularities

**ω**= [

*ω*,

_{x}*ω*,

_{y}*ω*]

_{z}*is the vector of angular velocity of the mobile platform. Therefore, the following matrix velocity equation can be written:*

^{T}**J**

^{–1}. Finally, to find all singularities, we calculate the determinant of that matrix:

*t*

_{1}

*t*

_{2}

*t*

_{3}= 0. Geometrically, this condition holds true when the three planes

*π*

_{1},

*π*

_{2}, and

*π*

_{3}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

*γ*

_{1},

*γ*

_{2}, and

*γ*

_{3}are <45°, so our mechanism is never at a singularity.

## 3. Static analyses

*t*

_{123}=

*t*

_{1}

*t*

_{2}

*t*

_{3}and

*d*

_{12}=

*d*

_{1}–

*d*

_{2}.

**τ**, and the wrench,

**F**, is

*x*, is resisted only by actuators 1 and 2. Furthermore, as the value of

*d*

_{12}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

*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).

*γ*

_{1},

*γ*

_{2}, and

*γ*

_{3}. However, as we will be interested in orienting a specific tool reference frame, not the mobile

*C*–

*x*′

*y*′

*z*′ reference frame, it is very difficult to discuss the orientation workspace of a general case.

*x*′

*y*′

*z*′ about an axis in the

*x*′

*y*′ 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 ).

*θ*from 0° until a mechanical interference is reached. The mechanical limits that we test for are angles

*γ*

_{1},

*γ*

_{2}, and

*γ*

_{3}(40°) and the actuator strokes (500 mm).

## 5. Alternative designs

*a*, 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. 7

*b*. This kind of architecture is useful for pick and place operations.

*a*, 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.

*b*, 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.

## 6. Conclusions

## References

**36**(1): 15–28.

**21**(9): 791–798.

**22**(10): 535–547.

*In*Advances in robot kinematics: motion in man and machine.

*Edited by*J. Lenarcic and M. Stanisic. Springer, Dordrecht, the Netherlands. pp. 211–221.

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*Transactions of the Canadian Society for Mechanical Engineering*.

**44**(4): 558-565. https://doi.org/10.1139/tcsme-2019-0189

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