# Mechatronic Model of a Compliant 3PRS Parallel Manipulator

^{*}

## Abstract

**:**

## 1. Introduction

_{z}flexure stage with nanometric accuracy is presented in [9], and a cartesian XYZ compliant mechanism is designed in [10]. Another high-performance three-axis serial-kinematic nano-positioning stage for high-bandwidth applications is developed in [11], and in [12], where a large range modular XYZ compliant parallel manipulator is presented, where the structure is composed by identical spatial double four-beam modules.

## 2. Mechatronic Model and Hypotheses

_{c}, θ

_{c}, p

_{zc}) is introduced into the inverse kinematic problem to obtain the commanded joint space coordinates (s

_{1c}, s

_{2c}, s

_{3c}). Second, these are introduced into the dynamic model of the control and actuators, which models the cascaded position, velocity and current control loops as well as the actuators dynamics. As the force required to deform the compliant mechanism can be higher than in a conventional mechanism, it is advisable to model the stiffness of the actuator and its transmission chain, as it may happen that a given transmission is less rigid than the compliant mechanism itself. For that purpose, in the present work, a 2-degrees-of-freedom model with two inertias linked by a torsional spring and damper is proposed. As a result of this modeling, the position reached by the actuators is obtained, which is, third, fed into the direct kinematic problem to know the simulated position of the manipulator.

_{1}, F

_{2}, F

_{3}) are then fed back into the model of the actuators, where they act as if they were disturbances that the control must counteract to reach the desired position. This approach of decoupling actuators and control dynamics from the manipulator dynamics allows a more detailed modeling of the control or actuator, as is the case here with 2-degrees-of-freedom modeling in state-space.

## 3. Kinematics

_{i}B

_{i}of equal length L connect the base and the mobile platform by means of an actuated prismatic joint P, a revolute joint R in C

_{i}, and a spherical joint S in B

_{i}. The angles α

_{i}between the bars and the fixed base are 45° in the default position. The B

_{i}points are in a circumference of radius b with center in P. The origin of the linear actuators is located in A

_{i}over a circumference of radius a and center in O. The actuators are inside three vertical planes at 120°, and their location is defined by the joint space coordinates s

_{i}.

_{1}, and the Z-axis is vertical. The XOY plane is horizontal at a height defined by the center of the compliant revolute joints. The mobile system UVW has the origin in P with the U-axis aligned with PB

_{1}and the W-axis perpendicular to the mobile platform. It is also shifted in such a way that the UPV plane contains the center of the compliant spherical joints.

_{x}, p

_{y}, and p

_{z}that determine the position of P and three angles ψ, θ, and ϕ, which are the rotations around the X-, Y- and Z axes. This is a low mobility parallel mechanism with 3 degrees of freedom p

_{z}, ψ, and θ, and 3 parasitic motions, p

_{x}, p

_{y}and ϕ. The parasitic motions of this 3PRS mechanism are already presented in [24] and here, for completeness, we have included the formulation in Appendix A. In addition, the inverse kinematics problem to calculate the joint space coordinates s

_{i}to reach a given position of the mobile platform in shown in Appendix B. The direct kinematics problem to determine the task space coordinates from the joint space ones is presented in Appendix C.

_{i}. The torsional rotation around the CiBi bars is measured by the angle β

_{li}, the bending rotation inside the vertical planes that contain the bar is determined by β

_{mi}, and the angle due to the bending in perpendicular direction to the vertical planes is β

_{ni}.

#### 3.1. Jacobians

#### 3.1.1. Jacobian of the Mobile Platform

_{i}, both through P and A

_{i}, the following equation is obtained:

**l**unity vectors define the direction of the bars and

_{i0}**s**define the positive direction of the actuators. The derivative with respect to time is

_{i0}**v**is the velocity of P,

_{p}**w**is the angular velocity of the mobile platform, and

_{p}**w**is the angular velocity of the bar i. Applying the dot product by

_{i}**l**,

_{i0}**m**Those restrictions can be expressed as follows:

_{i0.}#### 3.1.2. Jacobian of the B_{i}C_{i} Bars

**PB**and rearranging,

_{i}**w**can be posed as follows:

_{i}**J**is the Jacobians that relate the angular velocity of the bars with the velocity of the actuators:

_{αi}#### 3.1.3. Jacobian of the Rotation of the Spherical Joints

**l**and rearranging,

_{i0}_{j}, n

_{j}and l

_{j}in the local reference system S

_{j}, premultiplied by the corresponding rotation matrix

**R**as defined in Equation (23).

_{j}## 4. Dynamics

_{n}. The equations of motion of those elements are posed as a function of the generalized forces τ

_{n}and the Lagrangian L

_{n}, dependent on the kinetic energy T

_{n}and the potential energy V

_{n}.

**q**is a function of the global generalized coordinates of the assembled mechanism

_{N}**q**. Hence, their virtual displacements can be related through the corresponding Jacobian:

_{s}**q**=

_{s}**s**= {s

_{1}s

_{2}s

_{3}}

^{T}and

**τ**= { F

_{s}_{1}F

_{2}F

_{3}}

^{T}. This approach allows the direct and systematic obtainment of the forces on the manipulator, and it was partially developed in [11], where the moments due to the flexures’ deflection were considered external actions. Here, the complete formulation, including the proper modeling of the bending and torsional moments in the joints, is considered.

#### 4.1. Mobile Platform

**f**= {F

_{extm}_{u}F

_{v}F

_{w}}

^{T}is applied on a point D located as

**PD**= {d

_{m}_{u}d

_{v}d

_{w}}

^{T}, the force and the moments on the manipulator in the fixed reference system XYZ are

#### 4.1.1. Translational Dynamics

_{plat}is the mass of the mobile platform and g is the gravity. Applying the Lagrange equation, the equations of motion in matrix form are

#### 4.1.2. Rotational Dynamics

**J**Jacobian is applied. In addition, as the external torque is already calculated in the XYW reference system, it is premultiplied by the platform Jacobian.

_{q}#### 4.1.3. Contribution of the Elastic Energy in the Spherical Joints

_{sf}and k

_{st}are the flexural and torsional stiffnesses of the spherical joints, and

**β**is the angles obtained in Equation (26). Having calculated the corresponding Jacobians in Equation (28), the contribution to the equation of motion can be posed directly as

_{i}#### 4.2. Couplings between Actuators and 3PRS Manipulator

_{i}, where F

_{i}is the forces that the actuators must do. The Lagrangian takes into account the kinetic energy of the couplings and the elastic energy due to the deflection of the compliant revolute joints, where M

_{i}is the mass of the couplings and k

_{r}is the flexural stiffness of the joints. Being that α

_{0}is the α angle in the default position with no deformation,

#### 4.3. B_{i}C_{i} Bars

_{Gi}, y

_{Gi}, and z

_{Gi}, and their angle α

_{i}. Being that M

_{b}is the mass of the bars and I

_{bG}is their inertia moment in that point, their Lagrangian is

**J**, the contribution of each bar to the dynamics of the manipulator is

_{bi}#### 4.4. Dynamic Model of the Whole Mechanism

#### 4.5. Dynamic Model of the Control and Actuators

_{ic}in the joint space. Then, the position control is done using a cascaded control of position, velocity and current that is modelled in Simulink following the scheme of Figure 6. A proportional controller (k

_{V}) is used in the position loop, whereas a proportional–integral (k

_{P,}k

_{I}) control is used in the velocity loops. As the current loop runs at a lower loop cycle, the conversion from current to torque is assumed to be immediate through the motor torque constant k

_{T}, so the current loop is simplified in this way. The transmission ratio of the actuators is i

_{R}.

## 5. Experimental Validation

## 6. Discussion

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Parasitic Motions

_{i}in the mobile reference system is

**u**,

**v**and

**w**are the unity vectors that define the axis of the UVW reference system, and c and s refer to cosine and sine, respectively. Hence, the position of each spherical joint in the fixed reference system XYZ is

_{i}restrict the motion of the spherical joints to a plane defined by the linear actuators OA

_{i}and the manipulator bars C

_{i}B

_{i}, so the following conditions must be met:

## Appendix B. Inverse Kinematic Problem

**C**that locate the spherical joints with respect to the prismatic joints can be written as in Equation (A9). Their modulus is the length L of the bars:

_{i}B_{i}**s**define the positive direction of the actuators motion:

_{i0}**l**unity vector defines the direction of the bars and is calculated as

_{i0}_{i,}is calculated.

## Appendix C. Direct Kinematic Problem

_{i},

**C**bars in the following equations,

_{i}B_{i}**OC**vectors are

_{i}_{z}and the angles Ψ, and θ of rotation around the X and Y axes.

## Appendix D. Passive Coordinates

_{i}between the C

_{i}B

_{i}bars and the XY plane can be calculated as

_{i}and the default angle 45°.

_{mi}, β

_{ni}and β

_{li}, the relative rotation matrix between the MNL reference systems in Figure A1 must be obtained. Two reference systems MNL are used with the origin in the B

_{i}spherical joints. The first one, S

_{i0}, is fixed to the mobile platform, and the second one, S

_{i}, is fixed to the C

_{i}B

_{i}bars, as it is seen in Figure A1. Rotations around the axis defined by unity vectors

**m**and

_{i}**n**, β

_{i}_{mi}and β

_{ni}, refers to the deflections in the spherical joint in those directions, and rotation around the axis defined by

**l**, β

_{i}_{li}, measures the torsional deformation of the joint.

_{i0}system in the fixed reference system XYZ is,

**l**is already shown in Equation (A11). Additionally,

_{io}**m**is the unit vectors orthogonal to the vertical planes that contain the bars, defined as

_{io}**n**are calculated from the following cross product:

_{i0}**R**of the reference system S

_{i0}_{i0}is

**m**,

_{i}**n**and

_{i}**l**in the UVW system are defined as

_{i}**R**between reference systems S

_{i-i0}_{i0}and S

_{i}is obtained as

**R**can be developed as a function of the three consecutive rotations around m, n and l axis. The resultant matrix is

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**Figure 1.**(

**Top**) 3PRS compliant parallel manipulator developed. (

**Bottom**) Revolute and spherical joints used.

**Figure 2.**FEM simulations: (

**Left**) Maximum position in Z direction. (

**Right**) Maximum tilting around Y axis.

**Figure 8.**Experimental vs. simulated signals: (

**Left**) Angular position of the motors. (

**Right**) Motor torque.

**Figure 9.**Experimental vs. simulated signals: (

**Left**) Angular position of the motors. (

**Right**) Motor torque.

Parameter | Value | Units |
---|---|---|

a | 125.137 | mm |

b | 47.61 | mm |

L | 109.215 | mm |

M_{b} | 0.028 | kg |

I_{bG} | 2.36 × 10^{−5} | kgm^{2} |

M_{plat} | 0.153 | kg |

α_{0} | 45 | deg. |

I_{platu} | 6.834 × 10^{−5} | kgm^{2} |

I_{platv} | 6.834 × 10^{−5} | kgm^{2} |

I_{platw} | 1.309 × 10^{−5} | kgm^{2} |

M_{i} | 0.204 | kg |

k_{r} | 98.37 | Nm/rad |

k_{sf} | 32.665 | Nm/rad |

k_{st} | 24.46 | Nm/rad |

i_{R} | 2 × π × 14/0.07 | - |

k_{V} | 65 | 1/s |

k_{P} | 1 | As |

k_{I} | 8 | A |

k_{T} | 0.0302 | Nm/A |

J_{1} | 1.423 × 10^{−4} | kgm^{2} |

J_{2} | 4.817 × 10^{−8} | kgm^{2} |

c_{t} | 1.085 × 10^{−5} | Ns/m |

c_{1} | 0.003 | Ns/m |

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**MDPI and ACS Style**

Ruiz, A.; Campa, F.J.; Altuzarra, O.; Herrero, S.; Diez, M.
Mechatronic Model of a Compliant 3PRS Parallel Manipulator. *Robotics* **2022**, *11*, 4.
https://doi.org/10.3390/robotics11010004

**AMA Style**

Ruiz A, Campa FJ, Altuzarra O, Herrero S, Diez M.
Mechatronic Model of a Compliant 3PRS Parallel Manipulator. *Robotics*. 2022; 11(1):4.
https://doi.org/10.3390/robotics11010004

**Chicago/Turabian Style**

Ruiz, Antonio, Francisco J. Campa, Oscar Altuzarra, Saioa Herrero, and Mikel Diez.
2022. "Mechatronic Model of a Compliant 3PRS Parallel Manipulator" *Robotics* 11, no. 1: 4.
https://doi.org/10.3390/robotics11010004