A Robotic System with a Hybrid Motion Cueing Controller for Inertia Tensor Approximation in Micro-Manipulations

Abstract

This paper summarizes the development of a robotic system for the approximation of inertia tensor of micro-sized rigid bodies. We described the design and computer-based simulation of a 6-DOF motion platform in our earlier work [32] that benefits from an anthropological serial manipulator design. In [32] we emphasized that, in contrast to a standard configuration based on linear actuators, a mechanism with actuator design inspired from an anthropological kinematic structure offers relatively a larger motion envelope and higher dexterity making it a viable motion platform for micromanipulations. After having described the basic design and kinematic analysis of our motion platform in [32], we now aim to propose an advanced motion cueing algorithm for facilitating the identification of inertial parameters at micron-level. The motion cueing algorithm for achieving high fidelity dynamic simulation is described in this paper using a hybrid force-position-based controller. The inertia tensor identification is done by generating a controlled motion on the specimen and measuring the resultant forces and moments to approximate the inertia tensor using rigid body dynamics equations. The paper evaluates the performance of the controller using closed-loop dynamic simulations and validates the significance of the proposed method through experimental results.

Keywords

Parallel robot Micro-electromechanical Devices Real-Time Simulation SimMechanics,3D CAD Modeling,Dynamics Modeling,Closed-Loop Control

1. Introduction

In the field of motion dynamics, the identification and measurement of inertial parameters like center of mass and inertia tensor are generally accomplished through CAD models however, with the increase in complexity of the system, these standard methods suffer from uncertainty [1, 2 & 3]. The requirement for precise evaluation of inertial parameters of micro-sized objects is particularly important in the dynamic modeling and simulation of aircraft and spacecraft flight dynamics, vehicles, robots and many other automation processes.

During the last few decades, several methods [4–7] for identifying and determining rigid-body inertial properties have been investigated and presented. Traditionally, methods based on frequency response functions and high rotational speed experiments are conducted to identify the centre of mass and inertia tensor of rigid bodies [5]. These experiments require positioning and orientation of the body/assembly at high speed that may damage certain sophisticated assemblies or micro-sized components and therefore the problem of devising a robust solution to identify the inertial parameters of micro-sized objects is still under investigation.

Through recent research studies [8 & 44], it has been well understood that the uncertainty on the estimation of the inertial parameters can be reduced by using a complex motion that may involve both translational and rotational motions simultaneously. Therefore, the motivation of this paper is inspired from the work presented in [44] which delivers an excellent study with appreciable experimental results on the identification of inertial parameters of rigid bodies using a Stewart platform. However, in contrast to using a standard parallel robot with a Stewart-Gough configuration, we aim to propose an advanced motion cueing algorithm for our motion platform [32] capable of achieving micro-manipulations with adequate accuracy to aid in the inertia tensor measurements of micro-sized objects by means of tracking a controlled motion on the body and measuring the resultant forces and moments. A parameter identification procedure [9] can further be used to eventually determine the inertia tensor of the desired object using rigid body dynamics equations.

1.2 Micromanipulators

Micromanipulators have been developing widely because of its potential applications in fields such as optical fiber alignment, micro device assembly and micro-scale experiments in biological research [10]–[24]. Parallel mechanisms are robotic architectures that offer high dexterity, rigidity and precision when interacting with the environment. Therefore more and more micro robots are made of hybrid and parallel structures to acquire perfect performance. However, conventional parallel mechanisms feature fewer degrees of freedom [11]–[13] that restrict their motion envelope. Therefore, micromanipulators with 6-DOF motion capabilities with a reasonable workspace still require further research.

Shao-Chi Wang [20] dealt with the kinematics and dynamics analysis of a 6 degree-of-freedom RTS fully parallel manipulator with elastic joints. Dezhong Liu [21] developed an intelligent micro-manipulator based on 3-PRS in-parallel mechanism. A precision compliant parallel-structure positioner dually driven by six piezoelectric motors and six piezoelectric ceramics was designed by W. Dong. [22]. Based on the well-known tripod parallel configuration, design and modeling of a novel 3-DOF precision micro-stage was studied by Hwa Soo Kim [23]. Most of the 6-DOF parallel manipulators described in the literature contain six limbs [15, 16]. However, the more limbs the manipulator possesses, the more complicated that will be. A few of parallel manipulators employing compliant mechanisms have been designed to perform the manipulation in micro/nano meter scales with high accuracy, speed, and load capacity [13]–[28]. However, most of the existing micromanipulators can provide only a planar 3-DOF motion, or spatial 3-DOF combined motions of translation and rotation.

2. Design and Modeling

Some of the kinematic structures have linear motors at the base moving along a triangle as presented in [29] providing three DOFs, whereas 6-DOF utilizes rotational actuators as demonstrated in [30]. Another study describing a design with linear motion and a screw-like motion of the moving platform is shown in [31].

Based on these design considerations, we described a hybrid architecture resembling a parallel robot to achieve a reasonable workspace with minimum possible hinges and links in our previous work [32]. However the work was limited to 3D CAD model and a computer-based kinematic simulation only. Goal and contribution of this paper is to conclude the discussion opened in [32] and to present a practical application of our motion platform in the field of micro-manipulation technology. To do so, we aim to present a robotic solution for facilitating the identification of inertial parameters of rigid bodies in micromanipulations, by describing an advanced motion cueing algorithm based on a hybrid force-position-based controller to generate a controlled motion. The rest of this paper is structured as follows: the kinematic and dynamic models are described in sections 2.1 & 2.2 respectively, the control system incorporating a force-position-based controller is described in section 3, section 4 describes the identification procedure of inertial parameters such as inertia tensor, section 5 illustrates the significance and effectiveness of our proposed method through a comparative study of experimental results between the method proposed herein and others

Figure 1.

CAD model of the proposed motion platform.

found in literature. Finally, the discussion is concluded by narrating the advantages and shortcomings of our proposed method. Future goals are further identified in the conclusions section.

Figure 2.

Kinematic Configuration of the Proposed Architecture.

The architecture is composed of three sub-assemblies connected between the motion and fixed plates through passive and active joints. Figure 1 shows a variant of the design proposed in [32]. A motion plate (Body) represented by C₁C₂C₃ is connected to the bottom (fixed) plate H₁H₂H₃ through three subassemblies acting as legs to form a closed-loop parallel mechanism [31] as shown in Figure 2. Each leg consists of three links: link 1, link 2 & link 3 represented by lengths l₁, l₂ & l₃ inter-connected through two rigid rotational joints θ₂ & θ₃. Each subassembly (leg) is connected between the fixed plate and the motion plate through a rigid rotational joint represented by θ₁ and a flexible universal hinge respectively. Figure 2 describes the kinematic configuration of the proposed architecture. As shown in Figure 2, the fixed plate is aligned with the global coordinate system X^G−Y^G−Z^G whereas, the body coordinate system X^B−Y^B−Z^B moves with the motion of the Centre of Mass (COM) of the mechanism. Figure 3 shows different views of our prototype under development.

2.1 Kinematic Model

The goal of the kinematic model is to find the position of points C₁, C₂ & C₃ and the angular rotations of the revolute joints within each leg(actuator) for a given pose (specified by its position and orientation) of the motion plate. The kinematic model is considered here by determining the forward and inverse kinematic equations of a single actuator using Denavit-Hartenberg convention. The forward or direct kinematics computation of a single actuator connected between the fixed and motion plates yields the position of C_i in body frame {B} further described by Eq. 1.

\{\begin{matrix} {T_{B}^{B}}_{i} = {R^{B}}_{1} \times {R^{1}}_{2} \times {R^{2}}_{3} \\ {R^{B}}_{1} = [\begin{matrix} c o s θ_{1} & 0 & s i n θ_{1} & l_{1} \times c o s θ_{1} \\ s i n θ_{1} & 0 & - c o s θ_{1} & l_{1} \times s i n θ_{1} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ {R^{1}}_{2} = [\begin{matrix} c o s θ_{2} & - s i n θ_{2} & 0 & l_{2} \times c o s θ_{2} \\ s i n θ_{2} & c o s θ_{2} & 0 & l_{2} \times s i n θ_{2} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ {R^{2}}_{3} = [\begin{matrix} c o s θ_{3} & - s i n θ_{3} & 0 & l_{3} \times c o s θ_{3} \\ s i n θ_{3} & c o s θ_{3} & 0 & l_{3} \times s i n θ_{3} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ {T_{B}}_{x i}^{B} = c o s θ_{1} (l_{1} + l_{3} c o s θ_{23} + l_{2} c o s θ_{2}) \\ {T_{B}}_{z i}^{B} = s i n θ_{1} (l_{1} + l_{3} c o s θ_{23} + l_{2} c o s θ_{2}) \\ {T_{B}}_{y i}^{B} = {- l}_{3} s i n θ_{23} - l_{2} s i n θ_{2} \end{matrix}

(1)

Here, we impose the first requirement that Euler angles φ and ψ must describe exactly the direction of the y-axis with respect to the fixed plate frame of reference that is aligned with the global coordinate system. This requirement is adequately met by choosing the Euler angles (ψ, γ, φ) shown in Figure 4. In this orientation representation, the motion plate is first rotated about the base x-axis by an angle ψ, then about the new y-axis by an angle γ and finally about the z-axis by an angle φ. Thus, the position of B_i in global frame {G} is described by Eq. 2.

Figure 3.

Scaled prototype of our parallel robot under construction.

\{\begin{matrix} {T_{B}^{G}}_{i} = R_{z, φ} \times R_{y, γ} \times R_{x, ψ} \times [\begin{matrix} {T_{B}}_{x i}^{B} \\ {T_{B}}_{y i}^{B} \\ {T_{B}}_{z i}^{B} \end{matrix}] \\ R_{z, φ} = [\begin{matrix} c o s φ & - s i n φ & 0 & 0 \\ s i n φ & c o s φ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ R_{y, γ} = [\begin{matrix} c o s γ & 0 & s i n γ & 0 \\ 0 & 1 & 0 & 0 \\ - s i n γ & 0 & c o s γ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ R_{x, ψ} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & c o s ψ & - s i n ψ & 0 \\ 0 & s i n ψ & c o s ψ & 0 \\ 0 & 0 & 0 & 1 \end{matrix}] \\ {T_{B}^{G}}_{x i} = \cos γ \cos φ {T_{B}}_{x i}^{B} \\ - \cos γ \sin φ {T_{B}}_{y i}^{B} + \sin γ {T_{B}}_{z i}^{B} \\ {T_{B}^{G}}_{y i} = (\cos ψ \sin φ + \sin ψ \sin γ \cos φ) {T_{B}}_{x i}^{B} \\ + (\cos ψ \cos φ - \sin ψ \sin γ \sin φ) {T_{B}}_{y i}^{B} \\ - \sin ψ \cos γ {T_{B}}_{z i}^{B} \\ {T_{B}^{G}}_{z i} = (\sin ψ \sin φ - \cos ψ \sin γ \cos φ) {T_{B}}_{x i}^{B} \\ + (\sin ψ \cos φ + \cos ψ \sin γ \sin φ) {T_{B}}_{y i}^{B} \\ + \cos ψ \cos γ {T_{B}}_{z i}^{B} \end{matrix}

(2)

The general homogenous transformation equation is written as:

{T_{B}^{B}}_{i} = {R^{B}}_{1} \times {R^{1}}_{2} \times {R^{2}}_{3}

(3)

We rewrite the general homogenous transformation equation Eq. 3 of the actuator as Eq. 4 and compare r₁₄, r₂₄, r₃₄ elements of both sides. This yields joint angle θ₁ as described by the computations in Eq. 5.

i n v ({R^{B}}_{1}) \times [\begin{matrix} r_{11} & r_{12} & r_{13} & P_{x} \\ r_{21} & r_{22} & r_{23} & P_{y} \\ r_{31} & r_{32} & r_{33} & P_{z} \\ 0 & 0 & 0 & 1 \end{matrix}] = {R^{B}}_{2} \times {R^{B}}_{3}

(4)

\{\begin{matrix} {T_{B}}_{x i}^{B} \times s i n θ_{1} = {T_{B}}_{z i}^{B} \times c o s θ_{1} \\ \tan θ_{1} = \frac{{T_{B}}_{z i}^{B}}{{T_{B}}_{x i}^{B}} \\ θ_{1} = A T A N 2 ({T_{B}}_{z i}^{B}, {T_{B}}_{x i}^{B}) \end{matrix}

(5)

We rewrite the general homogenous transformation equation of the actuator as Eq. 6 and compare r₁₄, r₂₄, r₃₄ elements of both sides to yield Eq. 7, Eq. 8 7 Eq. 9.

i n v ({R^{B}}_{1}) \times i n v ({R^{B}}_{2}) \times [\begin{matrix} r_{11} & r_{12} & r_{13} & P_{x} \\ r_{21} & r_{22} & r_{23} & P_{y} \\ r_{31} & r_{32} & r_{33} & P_{z} \\ 0 & 0 & 0 & 1 \end{matrix}] = {R^{B}}_{3}

(6)

c o s θ_{2} \times ({T_{B}}_{x i}^{B} \times c o s θ_{1} - l_{1} + {T_{B}}_{y i}^{B} \times s i n θ_{1}) - l_{2} + {T_{B}}_{z i}^{B} \times s i n θ_{2} = l_{3} \times c o s θ_{3}

(7)

{T_{B}}_{z i}^{B} \times c o s θ_{2} - s i n θ_{2} \times ({T_{B}}_{x i}^{B} \times c o s θ_{1} - l_{1} + {T_{B}}_{y i}^{B} \times s i n θ_{1}) = l_{3} \times s i n θ_{3}

(8)

{T_{B}}_{x i}^{B} \times s i n θ_{1} - {T_{B}}_{y i}^{B} \times c o s θ_{1} = 0

(9)

Solving Eq.7, Eq. 8 & Eq. 9 yields θ₂ & θ₃ as described by Eq. 10.

\{\begin{matrix} θ_{2} = A T A N 2 (h_{i}, - {T_{B}}_{y i}^{B}) + \\ A T A N 2 ((\sqrt{{- {T_{B}}_{y i}^{B}}^{2} + {h_{i}}^{2} - {n_{i}}^{2}}), n_{i}) \\ θ_{3} = A T A N 2 ((- {T_{B}}_{y i}^{B} \cos θ_{2} - \sin θ_{2} h_{i}), \\ (h_{i} \cos θ_{2} - l_{2} - \sin θ_{2} {T_{B}}_{y i}^{B})) \\ h_{i} = ({T_{B}}_{x i}^{B} c o s θ_{1} - l_{1} + {T_{B}}_{z i}^{B} s i n θ_{1}) \\ n_{i} = \frac{{(h_{i}}^{2} + {l_{2}}^{2} + {T_{B}}_{y i}^{B}^{2} - {l_{3}}^{2})}{2 l_{2}} \end{matrix}

(10)

Figure 4.

Representation of the platform orientation using the pitch-yaw-roll Euler angles.

2.2 Dynamic Model

Our dynamical model is based on the famous Newton-Euler [33, 34] dynamic model which has been developed using the motion equations of rigid bodies in Cartesian space [35] in terms of position and time derivatives of the actuator joint angles in joint space with the general form described in Eq. 11.

\{\begin{matrix} τ = M (θ) \ddot{θ} + V (θ, \dot{θ}) + G (θ) \\ M = \sum_{i = 1}^{3} (J_{M i}^{T} m_{i} J_{M i} + \frac{1}{2} J_{R i}^{T} I_{i}^{0} J_{R i}) \\ V_{i j k} = \sum_{j = 1}^{n} \sum_{k = 1}^{n} (\frac{\partial M_{i j}}{\partial q_{k}} - \frac{1}{2} \frac{\partial M_{j k}}{\partial q_{i}}) \\ G_{i} = \sum_{j = 1}^{n} m_{j} g^{T} J_{M j}^{i} \end{matrix}

(11)

Where, τ is the force-torque vector, M is the n x n inertia matrix, V is the n x 1 vector of the centrigugal and Coriolis terms and G is the n x 1 vector of the gravity terms. We take the inverse of the Jacobean transpose and multiply it with Eq. 11 to yield Eq. 12

J^{- T} τ = J^{- T} M (θ) \ddot{θ} + J^{- T} V (θ, \dot{θ}) + J^{- T} G (θ)

(12)

Next, we develop a relationship between joint space and Cartesian acceleration as described in Eq. 13.

\{\begin{matrix} \dot{δ} = J \dot{θ} \\ \ddot{δ} = \dot{J} \dot{θ} + J \ddot{θ} \\ \ddot{θ} = J^{- 1} \ddot{δ} - J^{- 1} \dot{J} \dot{θ} \end{matrix}

(13)

Substituting Eq. 13 into Eq. 12 yields Eq. 14

J^{- T} τ = J^{- T} M (θ) (J^{- 1} \ddot{δ} - J^{- 1} \dot{J} \dot{θ}) + J^{- T} V (θ, \dot{θ}) + J^{- T} G (θ)

(14)

From Eq. 14, we derive the expressions for the terms in Cartesian dynamics as described in Eq. 15.

\{\begin{matrix} f = M_{δ} (θ) \ddot{δ} + V_{δ} (θ, \dot{θ}) + G_{δ} (θ) \\ M_{δ} (θ) = J^{- T} (θ) M (θ) J^{- 1} (θ) \\ V_{δ} (θ, \dot{θ}) = J^{- T} (θ) (V (θ, \dot{θ}) - M (θ) J^{- 1} (θ) \dot{J} (θ) \dot{θ}) \\ G_{δ} (θ) = J^{- T} (θ) G (θ) \end{matrix}

(15)

Where, f is the force-torque vector estimated on the motion centroid of the motion platform in {G} frame of reference and δ is an appropriate Cartesian vector representing position and orientation of the motion centroid in {G}. This force f acting on the motion centroid can be applied by the actuators at the joints by using the relationship described in Eq. 16.

τ = J^{T} (θ) f

(16)

Thus, the joint torques is expressed as Eq. 17

τ = J^{T} (θ) (M_{δ} (θ) \ddot{δ} + V_{δ} (θ, \dot{θ}) + G_{δ} (θ))

(17)

The control law is further formulated as described in Eq. 18.

\{\begin{matrix} \bar{τ} = {\ddot{θ}}_{d} + K_{v} \dot{e} + K_{p} e \\ e = θ_{d} - θ \\ \dot{e} = {\dot{θ}}_{d} - \dot{θ} \\ \ddot{e} + K_{v} \dot{e} + K_{p} e = 0 \end{matrix}

(18)

Figure 5.

A Block Diagram of the proposed Control System: hybrid Position-Force Controller

A practical implementation of this control model is achieved through the use of a dual-rate control system as shown in Figure 5. The figure describes a block diagram of a Cartesian-based Decoupling and Linearizing controller in which the dynamic parameters are written as functions of actuator positions and are updated at a rate (100Hz) slower than the servo rate because they need to be updated at a rate related only to how fast the motion plate is changing configuration (100Hz). This is appropriate because we desire a fast servo of approximately 500Hz or even higher to maximize disturbance rejection and stability. Figure 5 is a block diagram of our control system that describes the integration of position and force controllers with the actual system (plant). The blocks S and $\bar{S}$ are diagonal matrices (acting as simple switches) introduced in order to switch between the two controllers. The objective here is to formulate a control framework as, when a certain degree of freedom is under force control, position errors on that degree of freedom are estimated using the position control and when position control is in effect to track the control inputs, the force errors are compensated by the force controller in the background. The kinematics, dynamics and the Jacobeans are written in global frame of reference and thus the sensed forces and motions are likewise transformed into the global frame of reference {G}.

3. Simulation Model

The mathematical derivation for a simulation model is complex because of the complexity of parallel mechanisms [36, 37]. A problem of visualization occurs when only numerical results are produced during the simulation. Furthermore, the behavior of the mechanical joints and body of the designed motion platform as they operate in real is difficult to be visualized. Thus a solution using SimMechanics is suggested here.

SimMechanics software is a block diagram modeling environment for the engineering design and simulation of rigid body machines and their motions, using the standard Newtonian dynamics of forces and torques. It also presents the visual representation of the robot during the simulation [38, 39]. In this context, C. C. Ng, S. K. Ong and A. Y. C. Nee [38] analyzed 3-DOF micro Stewart platform using the SimMechanics. Lan Wang & etc. [40] analyzed Assistant Robotic Leg using the SimMechanics. Furthermore, the work done in [41–43] demonstrates the modeling and simulation of manipulators in macro-world using SimMechanics. In the light of these investigations, a SimMechanics model of the proposed motion platform is created using a CAD translation utility provided by SimMechanics.

Figure 6.

Physical Model of the Proposed Motion Platform in SimMechanics

The block diagram shown in Figure 6 is the Plant subsystem of the simulation model which describes the physical model of the proposed motion platform. The mechanical assembly is defined by rigid bodies with their mass properties, motions, kinematic constraints and coordinate systems. The actuating elements are the joint actuators which can be actuated by angular rotations determined by the kinematic model for a particular motion. The sensing elements constitute body and joint sensors to record the rotation angles of joints and pose of rigid bodies (positions and rotations). The visualization of the motion platform is further shown in Figure 7.

Figure 7.

Visualization of the SimMechancis Model in 3D Animation Viewer.

Figure 8 illustrates the kinematic architecture of an individual leg in terms of its rigid bodies, joints and their motion axes. Link 1 is connected to the fixed plate through a revolute joint, Link 2 is interconnected between Link 1 & Link 3 through a revolute joint and Link 3 is finally connected to the moving plate through a (custom joint replicating a universal hinge shown in Figure 6 & Figure 8.

Figure 8.

Kinematic configuration of an individual actuator

3.1 Verification of the Model

In order to realize the success of the controller and the verification of our dynamical model, closed loop simulations are carried out where joints and body sensors are used to provide joint rates, accelerations and torques as the control feedback. The controller is tested by tracking a reference control input in the form of positions and orientations of the motion plate in three dimensional spaces. As shown in Figure 9, the reference motion data is extracted from a flight dynamical model of a fixed wing aircraft (turbo-propeller Piper Jet) in form of angular rates and accelerations. The objective is to construct a reference data that consists of motions in multiple channels so that the motion platform can be simulated with mixed motions simultaneously. For instance, roll and pitch rates coupled with heave or surge accelerations. In order to achieve this, we have interfaced our motion cueing controller with a commercial flight simulator “X-Plane” and the flight data (rates & accelerations) is extracted from its flight dynamical model over a UDP protocol using Matlab S-fuctions.

Figure 9.

Overall Control framework representing the integration of our proposed control system, sensors and the Plant system.

3.1.1 Simulation Test 1 – Motions in Separate Channels

In the first simulation test, the motion platform is simulated in a single channel at a time, i.e. either linear translation or angular rotation in a single axis at a time. The objective of this simulation test is to determine the motion bounds of the motion plate and validate the success of our proposed controller in tracking a reference input.

The following figures deliver a graphical demonstration of the results of this simulation test. Figure 10 & Figure 11 graphically demonstrate the result of a pure ±25 degrees pitch and roll rotations around z & x axes respectively. Figure 12 shows the result of a pure ±50 degrees yaw rotation around y axis. Similarly, Figure 13, Figure 14 & Figure 15 show the results of ±5mm vertical, and ±7mm horizontal linear translations of the motion platform along y, x & z axes respectively.

Figure 10.

Visualization of a ±25 Degrees Pitch

Figure 11.

Visualization of a ±25 Degrees Roll

Figure 12.

Visualization of a ± 50 Degrees Yaw

Figure 13.

Visualization of a ±5μm Vertical Heave

Figure 14.

Visualization of a ±7μm Horizontal Sway

Figure 15.

Visualization of a ±7μm Horizontal Surge

3.1.2 Simulation Test 2 – Motions in Combined Channels

In the second test, the motion platform is simulated with an input data constituting of motions in combined channels simultaneously, i.e. motions constituting of both linear translations and angular rotations. The objective of this simulation test is to validate the success of the controller to track a complex motion.

The reference data is represented by the solid red lines in Figure 16 & Figure 18 that demonstrate a comparative representation between the reference motion and the tracked motion. As apparent from the figures, the tracked motion either matches or trails the reference motion that validates the capability of our controller to track a complex motion within its motion bounds. Figure 17 & Figure 19 show the error plots to present the estimated tracking errors.

Figure 16.

Orientation tracking of the motion plate

Figure 17.

Errors estimated in the orientation tracking.

Figure 18.

Position tracking of the motion plate.

Figure 19.

Errors estimated in the position tracking.

4. Identification of Inertia Tensor

The identification method involves approximation of inertia tensor of a micro-sized object. The idea is inspired from the work presented in [44] that is, to attach the object to a MEMS-based six-axis load cell (ATI Nano17) coupled with a MEMS-based IMU employed at the motion centroid of the moving platform using a rigid joint. A 3D motion trajectory is then generated on the object by simulating the motion platform with a reference control input (angular rates & linear accelerations) and the resultant forces and moments are estimated online using the sensors feedback data. The coupled IMU provides control feedback (angular rates & linear accelerations) to track the reference input. Identification of the inertia tensor is carried out with respect to the motion centroid of the motion plate. The mass of the object and its center of mass position are measured using static experiments as further investigated in [44]. The inertia tensor of an object I_o is described by Eq. 19.

\{\begin{matrix} I_{o} = [\begin{matrix} {I_{x x}}_{o} & {I_{x y}}_{o} & {I_{x z}}_{o} \\ {I_{y x}}_{o} & {I_{y y}}_{o} & {I_{y z}}_{o} \\ {I_{z x}}_{o} & {I_{y z}}_{o} & {I_{z z}}_{o} \end{matrix}] \\ I_{x x} = \int_{b} (y^{2} + z^{2}) d m \\ I_{y y} = \int_{b} (z^{2} + x^{2}) d m \\ I_{z z} = \int_{b} (x^{2} + y^{2}) d m \end{matrix}

(19)

The experiment is carried out in an inverse-dynamics closed-loop mode. Thus, the motion platform is actuated using joint torques computed by the dynamics engine using Eq. 17 for a reference motion trajectory (specified in terms of absolute position and orientation of the motion plate in global frame of reference {G}). The force controller plays the active role while the position controller estimates positioning errors. As a consequence, angular rates and linear accelerations are measured on the object using the IMU sensor feedback, and forces and moments are measured using the force-torque transducer.

The motion dynamics considered here uses the general force-moment equilibrium model as described in Eq. 20.

\{\begin{matrix} M_{i} = m {a_{b}}^{G}_{i} \times ({r_{b}}^{G} - {d_{b}}^{G}) - I {α_{b}}^{G}_{i} \\ {a_{b}}^{G}_{i} = ({α_{b}}^{G}_{i} - {ω_{b}}^{G} {ω_{b}}^{T}) ({r_{b}}^{G} - {d_{b}}^{G}) + {\ddot{d}}_{b}^{G} \\ {ω_{b}}^{G}_{i} = [\begin{matrix} \dot{φ} + \dot{ψ} s i n θ \\ \dot{θ} c o s φ - \dot{ψ} c o s θ s i n φ \\ \dot{θ} s i n φ + \dot{ψ} c o s θ c o s φ \end{matrix}] \\ {ω_{b}}^{G} {ω_{b}}^{T} = [\begin{matrix} {ω_{2}}^{2} + {ω_{3}}^{2} & - ω_{1} ω_{2} & - ω_{1} ω_{3} \\ - ω_{1} ω_{2} & {ω_{1}}^{2} + {ω_{3}}^{2} & - ω_{2} ω_{3} \\ - ω_{1} ω_{3} & - ω_{2} ω_{3} & {ω_{1}}^{2} + {ω_{2}}^{2} \end{matrix}] \\ {α_{b}}^{G}_{i} = \\ [\begin{matrix} c o s φ (\ddot{θ} + \dot{φ} \dot{ψ} s i n θ) + s i n φ (\ddot{ψ} s i n θ + \dot{θ} \dot{ψ} c o s θ - \dot{θ} \dot{φ}) \\ s i n φ (\ddot{θ} + \dot{φ} \dot{ψ} s i n θ) + c o s φ (- \ddot{ψ} s i n θ - \dot{θ} \dot{ψ} c o s θ + \dot{θ} \dot{φ}) \\ \ddot{φ} + \ddot{ψ} c o s θ - \dot{θ} \dot{ψ} s i n θ \end{matrix}] \\ {\ddot{d}}_{b}^{G}_{i} = \sum_{i = 1}^{n} {\dot{ω}}_{b}^{G}_{i} \times {d_{b}}^{G}_{i} \\ {\dot{ω}}_{b}^{G}_{i} = [\begin{matrix} 0 & - {\dot{ω}}_{3} & {\dot{ω}}_{2} \\ {\dot{ω}}_{3} & 0 & - {\dot{ω}}_{1} \\ - {\dot{ω}}_{2} & {\dot{ω}}_{1} & 0 \end{matrix}] \end{matrix}

(20)

Where, M_i is the resultant moment vector estimated on the object using the force-torque sensor, a_b^G_i & α_b^G_i are the linear acceleration and angular rate vectors estimated using the IMU feedback and the outputs of our dynamical model in {G} frame. (r_b^G − d_b^G) is the position vector from the center of mass of the object to the motion centroid of the moving plate and I is the inertia tensor of the object to be approximated. Here, we devise a method of determining the components of inertia tensor using the Pseudo inertia matrix [33] as described in Eq. 21. Thus, by estimating the linear accelerations, angular rates, forces and moments on the object, the inertia tensor can be approximated with a method of least squares using the extended system of expressions as described in Eq. 21.

\{\begin{matrix} {\bar{I}}_{o} = (m {a_{b}}^{G}_{i} \times ({r_{b}}^{G} - {d_{b}}^{G}) - M_{i}) \times {α_{b}}^{G}_{i}^{- 1} \\ {\bar{I}}_{o} = [\begin{matrix} \int_{o} x^{2} d m & \int_{o} x y d m & \int_{o} x z d m & \int_{o} x d m \\ \int_{o} y x d m & \int_{o} y^{2} d m & \int_{o} y z d m & \int_{o} y d m \\ \int_{o} z x d m & \int_{o} z y d m & \int_{o} z^{2} d m & \int_{o} z d m \\ \int_{o} x d m & \int_{o} y d m & \int_{o} z d m & \int_{o} d m \end{matrix}] \\ {\bar{I}}_{o} = [\begin{matrix} \frac{{- I}_{x x}_{o} + {I_{y y}}_{o} + {I_{z z}}_{o}}{2} & {I_{x y}}_{o} & {I_{x z}}_{o} & m x_{c} \\ {I_{y x}}_{o} & \frac{{I_{x x}}_{o} - {I_{y y}}_{o} + {I_{z z}}_{o}}{2} & {I_{y z}}_{o} & m y_{c} \\ {I_{z x}}_{o} & {I_{y z}}_{o} & \frac{{I_{x x}}_{o} + {I_{y y}}_{o} - {I_{z z}}_{o}}{2} & m z_{c} \\ m x_{c} & m y_{c} & m z_{c} & m \end{matrix}] \end{matrix}

(21)

5. Experimental Results

To assess the performance and accuracy of the proposed method, some tests were conducted in which the inertia tensor of a micro-sized simple rigid body was measured, thus allowing comparison between the analytical and experimental results. The forces and moments are estimated using the six voltage signals generated by the force-torque transducer (ATI Nano 17).

Figure 20.

A comparative representation of measured and computed angular rates on the object.

Figure 21.

A comparative representation of measured and computed angular accelerations on the object.

Figure 22.

Torques measured on the object.

Figure 23.

Forces measured on the object.

A parallelepiped object of mass 0.008kg was fixed on the plate and tests were carried out with twenty different orientations of the object. The estimated angular rates and accelerations on the object are shown in Figure 20 & Figure 21 respectively. The forces and moments estimated on the object are shown in Figure 22 & Figure 23 while the results for a single orientation are listed in Table 1. The approximated inertia tensor results are listed in Table 2 compared with the analytical values obtained from rigid-body dynamics model. As apparent from the tabulated results, the inertia moments and product of inertia are approximated with good accuracy. Table 3 compares the accuracy of our proposed method with those of other methods [8], [9] & [10] and signifies the proposed method yielding lower errors.

Table 1.

Approximated Values of Forces and Moments

	F₁(N)	F₂(N)	F₃(N)	M₁(Nmm)	M₂(Nmm)	M₃(Nmm)
Expected Value	15.23	30.61	25.64	1.32	5.65	6.96
Estimated value	15.01	30.23	25.11	1.30	5.06	6.21

Table 2.

Results of inertia tensor approximation

	Inertia moments (kgmm²)			Inertia products (kgmm²)			Principle moments (kgmm²)
	I _xx	I _yy	I _zz	I _xy	I _yz	I _xz	I ₁	I ₂	I ₃
Analytic	8.34×10⁻⁵	7.98×10⁻⁵	1.34×10⁻⁵	1.12×10⁻⁵	2.12×10⁻⁸	1.67×10⁻⁶	3.43×10⁻⁵	4.11×10⁻⁵	6.45×10⁻⁵
Approximated	1.98×10⁻⁴	5.87×10⁻⁵	1.03×10⁻³	4.49×10⁻⁵	1.77×10⁻⁸	9.5×10⁻⁶	5.67×10⁻⁵	3.01×10⁻⁵	7.10×10⁻⁵
Error	1.15×10⁻⁴	2.11×10⁻⁵	1.37×10⁻³	5.61×10⁻³	3.89×10⁻⁴	2.62×10⁻⁵	9.10×10⁻⁵	1.11×10⁻²	7.16×10⁻³

Table 3.

Comparison with other approximation methods

	Frequency Response Method	Time Domain Model	Proposed Method
I_xx%	5.6	12	8.8
I_yy%	4.9	8.6	2.1
I_zz%	17	4.5	3.6

5. Conclusions and Future Work

In this paper and its companion [32], we presented a study on the design and practical implementation of a six-DOF motion platform inspired from an anthropological serial manipulator design in facilitating the identification of inertial parameters of micro-sized rigid bodies. In particular, the design and kinematic architecture of our motion platform was addressed in our earlier work [32], while in this part, we focused on its application in micromanipulations by performing a functional evaluation of inertial parameters such as inertia tensor of rigid bodies.

The method proposed herein for approximating the inertia tensor of micro-sized rigid bodies is quite fast since it requires accurate positioning of the object only and allows the determination of its inertia tensor with appreciable accuracy. The proposed method uses a closed-loop dynamic simulation model which reduces the experimentation time and the efforts consumed in post-processing, in comparison to traditional methods.

It is investigated that the uncertainty on the inertia tensor approximation can be reduced by generating a controlled motion on the object. To achieve this feat, a new architecture of a 6-DOF motion platform is presented in this paper. The architecture consists of only revolute hinges as connecting elements, which are already well established in MEMS. The kinematic configuration constitutes a biologically-inspired actuator model instead of traditional linear actuators which suffer from constrained work space. As a consequence, the proposed kinematic configuration consists of fewer links i.e. only 9 links to form a closed loop kinematic chain with motion capabilities of a parallel robot. The inverse kinematic solution is given along with the direct kinematics to describe the pose of the motion plate in global frame of reference. Instead of relying on the kinematic model only as we did in [32], in this paper we determined a complete dynamic model of our motion platform to bring the motion dynamics into consideration. Through closed-loop dynamic simulations it is further validated that the proposed architecture achieves a reasonable workspace both in single and combined motions. In single motions it achieves ±25 degrees of roll & pitch and ±50 degrees of yaw rotations. Similarly, ±5μm vertical and ±7μm horizontal motion bounds have been verified in linear translations in single channels.

The objective of generating a controlled motion on the specimen under measurement is achieved by integrating a hybrid force-position controller (Cartesian-based Decoupling and Linearizing) with the actual system and onboard sensors to form a hardware-in-loop simulation model. As a consequence, the control system is capable of tracking a complex motion specified in terms of angular rates and linear accelerations (taken from a flight dynamical model) with a capability of generating controlled motion that is within motion envelope of the motion platform to avoid singularities that are characteristic of parallel mechanisms.

Finally, the paper describes the approximation of the components of inertia tensor with the help of the Pseudo inertia matrix using the measured and estimated dynamics, achieved through the proposed controller. The paper further signifies the accuracy and performance of the proposed method through experimental results and comparison with analytical values and results drawn from other methods found in literature. Future development of this research will be the identification and approximation of center of mass of micro-sized objects using a similar approach with improved sensor fusion and disturbance rejection techniques. The development of an actual prototype is under process.

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