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A 3D-printed 3-DOF tripedal microrobotic platform for unconstrained and omnidirectional sample positioning

  • Iman Adibnazari
  • William S. Nagel
  • Kam K. Leang
Regular Paper
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Abstract

This paper describes the design of a 3D-printed, three-degree-of-freedom (3-DOF) tripedal microrobotic platform that is capable of unlimited travel with sub-micrometer precision over a planar surface. The design combines piezoelectric stack actuators with compliant mechanical amplifiers to create stick-slip-style mechanisms for locomotion. A forward kinematic model of the stage’s motion is derived from its tripedal leg architecture. The model is then inverted for feedforward control of the platform. A prototype of the microrobot is constructed using low-cost 3D-printing technology. Experimental results demonstrate actuator stroke of 29.4 \(\upmu\)m on average with a dominant resonance of approximately 860 Hz. Using open-loop feedforward control, the stage travels along a 3 mm \(\times\) 3 mm, rectangular path. Feedback control through visual servoing is then simulated on a model that includes flexure dynamics, observed surface interactions, and camera sampling times, reducing the root-mean-square (RMS) tracking error by 90%. This controller is then implemented experimentally, resulting in 99% RMS position error reduction relative to feedforward-only control structure. The results show feasibility of creating functional 3D-printed, 3-DOF sample positioning stages.

Keywords

Compliant mechanisms Microrobotic platform Piezoelectric actuators Stick-slip motion Visual servoing 

1 Introduction

Precision mechatronic stages provide the fields of imaging and optics unparalleled control over sample positioning, allowing for manipulation as precise as the sub-nanometer scale. These stages provide a means of automating sample positioning in applications including biological imaging (Yang et al. 2016), transmission electron microscopy (TEM) (Holuba et al. 2006), and atomic force microscopy (AFM) (Fleming and Leang 2014), overcoming the need for manual manipulation when dynamically analyzing samples. Among precision positioning techniques, piezoelectric stick-slip (PSS) actuators are a predominately standardized and robust method for manipulating samples with both high speed and fine resolution (Oubellil et al. 2015; Shiratori et al. 2012; Lu et al. 2015; Liang et al. 2017; Klingner et al. 2014; Shrikanth et al. 2015). In most applications however, the maximum range of PSS positioning stages will inevitably need to be accounted for and may prove restrictive. Furthermore, the fabrication process necessary for such high precision stages requires both expensive and generally inaccessible manufacturing techniques. This paper presents the development of a novel, tripedal, three-degree-of-freedom (3-DOF) robotic platform with submicrometer stepping resolution (henceforth referred to as a “microrobotic” platform) and unlimited range of motion on a planar surface, designed to be manufactured from polyactic acid (PLA) via commercially available 3D-printers. A conceptual illustration of the positioning stage is provided in Fig. 1, where the 3-DOF stage is shown in use underneath an optical microscope. The contributions of this work include: (1) the incorporation of mechanical amplifying flexure mechanisms for increased actuator stroke and resonance, (2) a kinematic model relating the inputs of the tripedal PSS architecture to the total platform translations and rotations, (3) experimental validation of both the platform’s constituent mechanisms, as well as the developed kinematic models, on a physical prototype, and (4) refinement of the microrobot’s trajectory tracking capabilities though feedback control, in particular via visual servoing, in both simulation and experimental testing.
Fig. 1

Concept of sample positioning stage for microscopy: a A digital microscope observing b a sample mounted to c the proposed sample positioning stage as it translates and rotates beneath the microscope. The design offers two degrees of translational freedom and one degree of rotational freedom

The ubiquity of PSS actuators stems from their capacity to operate in both a high-speed stepping mode and a high-resolution scanning mode. Both modes are based in leveraging frictional nonlinearities between a manipulator and a target surface (Lu et al. 2015; Liang et al. 2017; Klingner et al. 2014; Shrikanth et al. 2015). To achieve stepping mode operation, the PSS manipulator first deforms slowly so as to not overcome the coulomb friction that keeps the target surface “sticking” to the manipulator. The manipulator then quickly deforms back to its original state, overcoming coulomb friction between it and the target object. This allows the manipulator and target to “slip” relative to each other, resulting in the target object’s displacement. In scanning mode, the PSS manipulator only deforms slowly, maintaining its frictional hold on the target object and allowing for higher precision in manipulation. The use of PSS actuators in imaging stage designs has become commonplace, with novel architectures rarely developed (Shiratori et al. 2012); many stages utilize commercially available PSS actuators to achieve sample manipulation in up to six degrees-of-freedom (Lu et al. 2015). Applications of these PSS actuators in imaging utilized include TEM (Holuba et al. 2006) and optic alignment systems (Guyenot and Siebenhaar 1999). Stick-slip dynamics also play a dominant role in microrobot locomotion, where piezoelectric elements are a preferred method of actuation for small-scale robotic platforms (Qu et al. 2017; Murthy et al. 2008; Li et al. 2011; Rios et al. 2017).

The work presented in this paper focuses on the design and implementation of a 3D-printed, microrobotic platform that incorporates three PSS actuators in a tripedal arrangement to achieve 3-DOF mobility with unlimited range of motion and sub-micrometer precision. Mechanical flexures are included in the robot architecture to increase PSS actuator ranges such that the platform can move with step sizes up to approximately 10 \(\upmu\)m and with first resonances greater than 500 Hz for each actuator. These design goals allow the platform to mobilize microscopy samples with both sub-micrometer precision and up to speeds of 0.5 mm/s, making it suitable for a wide array of imaging and microscopy applications, while still remaining within the material limitations posed by 3D-printing with PLA. Comparable tripedal arrangements of actuators are employed in robots like the Droplet swarm robotic platform—developed by Klingner et al. (2014)—and the Abelone microrobot–developed by Zesch et al. (1995). However, the imprecision inherent to the Droplet’s motorized power train renders it impractical for imaging, and the precision required to manufacture the Abelone’s drive mechanisms results in a vastly more expensive and inaccessible platform than a comparable design explicitly selected for production via 3D-printing with readily accessible materials.

The proposed platform will be introduced in the following section along with its operating principle, the design and simulation of its constituent mechanisms, and kinematic models describing its motion. Afterwards, a prototype platform’s design and fabrication are presented along with verification of its mechanisms and open-loop trajectory tracking. The resulting data are then integrated into a model of the robot’s dynamics, which is then used to simulate feedback control via visual servoing through an optical microscope. The proposed control scheme is then experimentally verified on the prototype system. Finally, concluding remarks and future works are presented.

2 Microrobotic platform design

2.1 Operating principle: stick-slip motion

Microrobotic mobile platforms that employ PSS actuators in their legs treat the ground as the target that the actuators operate on. By arranging PSS actuators in a tripedal architecture, the base can be made capable of mobility in three degrees-of-freedom on a surface, allowing it to take steps of arbitrary translational and rotational motion, which can be seen in Fig. 2.
Fig. 2

Tripedal mobile base iterating through stick-slip gait to achieve pure rotation. Each stick-slip actuator is modeled by mass \(m_{i}\) and piezoelectric actuator, \(a_{i}\), with internal stiffness, \(k_{i}\), and damping, \(b_{i}\). Over the course of one step-cycle, a the legs are first unactuated; b the actuators then lengthen slowly to avoid overcoming coulomb friction, leaving the legs stuck to surface; and c the actuators then contract quickly to overcome coulomb friction and allow the legs to slip across the operating surface. Steps resulting in any combination of rotation and translation can be achieved by controlling the relative flexion of each foot during the stick phase of motion

Fig. 3

a Top view of fully assembled microrobotic platform; b1 top, and b2 side views of the microrobot after top cover is installed; c1 unactuated and c2 actuated monolithic flexures

Over the course of each step-cycle in stepping-mode, the proposed platform iterates through three phases of motion. First, shown in Fig. 2a, the legs are left at rest, allowing transients from previous step-cycles to decay. Next, the PSS actuators flex slowly such that the forces applied to the feet do not exceed the force of coulomb friction between the feet and ground. This causes the feet to “stick” in place while the body shifts with a desired displacement, depicted in Fig. 2b. Finally, the legs actuate to exert a force on the base that exceeds the coulomb friction between the feet and ground. This allows the feet to “slip” to where they had started relative to the body, as in Fig. 2c. After completing these phases, the base is left both displaced and in its resting state, allowing it to iterate through further step-cycles.

2.2 Platform overview

The proposed platform is a tripedal, monolithic microrobotic stage that utilizes flexure mechanisms to achieve stick-slip motion at the sub-micrometer scale (Fig. 3). With microscopy samples secured to its top surface, this platform embodies a novel paradigm for imaging stages: rather than a sedentary instrument mobilizing a target sample over a fixed stage with limited range, the entirety of the platform moves with the sample about a surface.

A primary objective of the platform is to offer a low-cost and accessible imaging stage. Wire electrical discharge machining (EDM) is the most common method of fabricating precision flexure mechanisms like those used in the proposed platform (Fleming and Leang 2014). Producing only the body of proposed platform through this method can be expected to cost well beyond $1000.00 (US). However, the presented platform is designed to be manufactured via 3D-printing and assembled using commercially available hardware and electronics. By using this manufacturing technique, the robot body, again excluding additional electronics and drivers, can be expected to cost on the order of $5.00 (US).

2.3 Mechanism design

A monolithic flexure mechanism, shown in Fig. 3c, is iteratively designed both to effectively amplify input displacements from a piezoelectric element (Lobontiu 2002). The increased displacement offered by the mechanical amplifier allows for an equivalently increased range of step sizes from the platform. Accordion guide-flexures are also added from each foot to the platform’s base both to decrease the effects of each foot’s resonance as well as to increase the maximum load carrying capacity of the platform by increasing the out-of-plane stiffness of each flexure.

As the platform’s mechanisms are entirely 3D-printed, an adequate simulation of the mechanisms’ behavior with regard to their material properties is developed. The platform is assumed to behave equivalently to a transversely isotropic sample of polylactic acid (PLA), with the plane of isotropy aligned with the platform’s top face (Jamshidian et al. 2010). Finite element analyses (FEAs) are conducted on the platform’s flexures to estimate their operating capabilities, and the design is adapted to jointly improve the flexure’s maximum stroke for a given piezoelectric actuator (ThorLabs PK3CMP1), mechanical amplification of input displacements, and resonant modes. The simulated flexure exhibits a maximum stroke of 33.58 \(\upmu\)m, an amplification factor of 3.62 \(\upmu\)m/\(\upmu\)m, and a first resonant mode at 668.3 Hz in the plane of actuation. The first resonant mode shape is in the direction of actuation with stresses and strains predominantly concentrated at points where the flexures join with the robot’s body and feet.

In the microrobot’s base, three piezoelectric stack actuators are employed to actuate three of the aforementioned monolithic flexures. These flexures take input displacements from their respective piezoelectric elements and transmit an amplified displacement to the robot’s feet.

An FEA is performed while the platform bares a distributed load of 1 kg. This analysis gives grounds to the assumption that if the flexures’ out-of-plane stiffness is high enough to allow for nearly ideal operation while bearing a 1 kg load, then the relative loading from a microscopy sample can be considered negligible to normal operation. Under this prescribed loading, the platform’s flexures maintain 95.6% of their unloaded stroke range, proving sufficient for the assumption that when loaded, the flexures will operate with their designed stroke characteristics.

2.4 Kinematic models

As mentioned above, the base of the platform contains three legs, each comprised of one piezoelectric actuator and one flexure mechanism. The arrangement of these actuators combined with their respective actuation directions allows for 3-DOF mobility on a plane, which can be seen in Fig. 4. Each piezoelectric actuator is preloaded and given a bias voltage when operated that allows for both compressive and tensile actuation. A kinematic model of the robot’s motion is developed to relate the flexions of each leg of the platform (\(l_{1}\), \(l_{2}\), and \(l_{3}\)) during a stick-slip cycle to the resulting displacement of the microrobot body. Two simplifying assumptions are made in developing this kinematic model:
  1. 1.

    During each stick-slip cycle, all legs experience perfect stick and slip simultaneously.

     
  2. 2.

    The final displacement of the platform is linearly related to the flexion achieved by each leg during the stick phase.

     
Assumption (1) is simple to achieve given a sufficiently fast electrical system providing voltage trajectories to each of the legs. Assumption (2) is made to decouple the effects of the legs upon one another as they flex during the stick phase and allows for the derivation of a linear map between flexions in the legs’ reference frames and a resulting displacement. Coupling between legs is otherwise expected in that, during the stick phase of a step-cycle, as one leg flexes and maintains its frictional hold on the operating surface, that leg will induce forces on the other legs. These forces unintentionally work to overcome the coulomb friction maintaining the other legs’ frictional holds on the operating surface. This coupling is partially reduced by the compliance of the platform’s body and mechanisms, but is present nonetheless.
Together, these assumptions allow for the following discrete kinematic model in the robot’s reference frame:
$$\begin{aligned} \mathbf {d_{r}}[k+1] = \varvec{T}~\mathbf {l_{r}}[k] \end{aligned}$$
(1)
where
$$\begin{aligned} \mathbf {d_{r}}[k+1]= \begin{bmatrix} x_{r}[k+1] \\ y_{r}[k+1] \\ \theta _{r}[k+1] \end{bmatrix} \end{aligned}$$
(2)
is the displacement vector resulting after the stick-slip cycle k,
$$\begin{aligned} \mathbf {l_{r}}[k]= \begin{bmatrix} l_{1}[k] \\ l_{2}[k] \\ l_{3}[k] \end{bmatrix} \end{aligned}$$
(3)
is a vector containing the leg flexions during stick-slip cycle k, and
$$\begin{aligned} \varvec{T} = \begin{bmatrix} -\cos (30^{\circ })&0&\cos (30^{\circ }) \\ \sin (30^{\circ })&-1&\sin (30^{\circ }) \\ \frac{\cos (30^{\circ })}{3d}&\frac{\cos (30^{\circ })}{3d}&\frac{\cos (30^{\circ })}{3d} \\ \end{bmatrix} \end{aligned}$$
(4)
is the transformation relating the leg flexions to the resulting displacement in the microrobotic platform’s frame of reference. In this matrix, the parameter d corresponds to the distance from the platform’s geometric center to the center of each foot that makes contact with the operating surface. Because \(\varvec{T}\) is full rank, the forward kinematic model can be inverted and used to derive the leg flexions necessary for a given displacement:
$$\begin{aligned} \mathbf {l_{r}}[k]= \varvec{T}^{-1} \mathbf {d_{r}}[k+1]. \end{aligned}$$
(5)
Stick-slip gaits other than that described in Assumption (1) can result in equivalent kinematic models. However, the derivation of \(\varvec{T}\) most clearly follows from the assumed gait. In a global reference frame, the forward kinematic model can be expanded to provide the discrete, nonlinear state equations
$$\begin{aligned} x_{g}[k+1] &= x_{g}[k] + x_{eff}[k]~\cos (\theta _{g}[k]) \\ &\quad -y_{eff}[k]~\sin (\theta _{g}[k]), \end{aligned}$$
(6)
$$\begin{aligned} y_{g}[k+1] &= y_{g}[k]+y_{eff}[k]~\cos (\theta _{g}[k]) \\& \quad +x_{eff}[k]~\sin (\theta _{g}[k]), \end{aligned}$$
(7)
and
$$\begin{aligned} \theta _{g}[k+1] = \theta _{g}[k]+\frac{\cos (30^{\circ })}{3d}~(l_{1}[k]+l_{2}[k]+l_{3}[k]), \end{aligned}$$
(8)
where \(x_{g}\), \(y_{g}\), and \(\theta _{g}\) correspond to the platforms position and orientation in the global reference frame, as shown in Fig. 5. The effort put forth by the PSS actuators to move in the \(x_{g}\) direction during stick-slip cycle k (that is, \(x_{eff}[k]\)) is defined as,
$$\begin{aligned} x_{eff}[k] = -\cos (30^{\circ })~l_{1}[k] + \cos (30^{\circ })~l_{3}[k], \end{aligned}$$
(9)
and \(y_{eff}[k]\) is the effort put forth by the PSS actuators to move in the \(y_{g}\) direction during stick-slip cycle k, defined to be
$$\begin{aligned} y_{eff}[k] = \sin (30^{\circ })~l_{1}[k]-l_{2}[k] + \sin (30^{\circ })~l_{3}[k]. \end{aligned}$$
(10)
Fig. 4

Robot leg architecture with parameters required to describe kinematic model in platform’s reference frame

Fig. 5

Local and global reference frames describing the platform’s position during stick-slip cycle k

3 Experiments and results

3.1 Manufacturing

The positioning platform is additively manufactured with PLA (eSUN 3 mm Black PLA PRO) using a commercial 3D printer (LulzBot TAZ 6) and 100% material infill to best achieve homogeneous material properties. Piezostack actuators (ThorLabs PK3CMP1) are installed and preloaded with finely threaded bushings (ThorLabs N100B2P and F19SSN1P). A 5.0 mm sapphire hemisphere (Edmund Optics #48-432) is glued to the output platform of each flexure mechanism, providing highly resilient point contact with surfaces. Access channels for piezostack preloading and capacitive sensors (ADE Technologies model 2823) installation are included in the final design. Lastly, hardware for mounting sample slides are set on the top face of the microrobot. These features can all be seen in Fig. 6.
Fig. 6

Microrobotic positioning platform prototype. The constituent elements of the platform are shown in the a top view and b bottom view

A serial data acquisition device (Zeltom SIMLAB) is used in conjunction with Piezodrive PDu100, 27.5 V/V amplifiers and signal biasing circuitry to drive the platform’s piezoelectric stick-slip actuators. Control signals are computed and sent to the electrical drivers using MathWorks’s Simulink Real-Time 2017b, the MATLAB Control Systems Toolbox, and the MATLAB Image Acquisition Toolbox.

3.2 Mechanism characterization

Characterization at the mechanism level is conducted to integrate into models of the microrobotic platform’s dynamics. Low-frequency stroke characteristics of the flexure are recorded via laser vibrometer (Polytech CLV 700) to determine the maximum achievable stroke from the flexures during the stick phase of each step. These measurements are reported in Table 1.
Table 1

Flexure characteristics

Parameter

Leg 1

Leg 2

Leg 3

Maximum stroke at foot (\(\upmu\)m)

28.43

29.4

30.41

First resonance (Hz)

836.3

867.7

875.4

The frequency response of each flexure is measured and shown in Fig. 7, where the input voltage to the system’s electrical driver is related to the output displacement of each flexure in micrometers. The flexure displacements are measured using capacitive sensors. An HP 35665A dynamic signal analyzer is used in the swept sine mode to measure the frequency responses. The first resonant frequency proved consistent between all three flexures, with precise values cataloged in Table 1.

The discrepancies between the legs’ frequency responses is attributed to the platform’s manufacturing process. While 100% infill is used during 3D-printing to best approximate material isotropy, the feet are not printed symmetrically and their precision is limited to the tolerance of the 3D-printer. Together, these manufacturing differences result in inconsistencies between frequency responses.

3.3 Locomotion

The platform’s step resolution is dependent on the frictional coefficients between its feet and the operating surface (Li et al. 2011; Eigoli and Vossoughi 2010). For sake of experimental consistency, the platform is operated on a sheet of 1008 steel that is finished with Kynar 500 coating. On this surface, the platform exhibited a maximum step-size of 5.88 \(\upmu\)m/step while operating at 100 steps/s, allowing it to achieve the proposed 0.5 mm/s movement speed.

The kinematic model is used in an open-loop control law to determine the platform’s locomotive ability while following paths to demonstrate 3-DOF motion. In tracking a rectangular path, the platform is to translate across planned squares with side lengths of 0.3 cm without rotating. Figure 8 shows the position of the platform as it tracks the rectangle at different speeds over multiple trials and the resulting position and orientation errors are reported in Table 2. A purely rotational path is also tracked, consisting of a \(20^{\circ }\) counterclockwise rotation followed by a \(20^{\circ }\) clockwise rotation, returning the platform to the original orientation. The resulting position and orientation errors of this trajectory are again presented in Table 2.

Figure 8 demonstrates the stage’s capacity to change translational direction instantaneously as well as the potential error induced while tracking trajectories at varying speeds. It is worth noting that the tracking error accumulated over the trajectory is partially attributable to platform’s lack of internal odometry for feedback control; thus, the platform depends entirely on feedforward control, based on inverting the kinematic model presented in the preceding section. These outlined negative effects can be mitigated in applications of the platform where an imaging device is used to analyze samples. In such applications, the position and orientation of the microrobot can be observed directly and used for closed-loop control.
Fig. 7

Experimentally measured frequency responses of flexure mechanisms in robot’s legs. Modeled frequency response is superimposed in the dashed line

Fig. 8

Position of positioning platform as it tracks a square with side lengths of 0.3 cm at speeds of 50 steps/s and 100 steps/s

Table 2

Induced error from open-loop trajectory tracking

Trajectory type

Operation speed (steps/s)

Position error (X, Y) (cm)

Orientation error

0.3 cm square

50

(− 0.172, 0.157)

6.470°

0.3 cm square

100

(− 0.195, 0.080)

6.911°

\(20^{\circ }\) arc

50

(− 0.0848, 0.4977)

− 1.998°

\(20^{\circ }\) arc

100

(− 0.1486, 0.3597)

− 3.628°

3.4 Closed-loop control simulation

The tracking error shown in Fig. 8 originates from multiple unmodeled effects on the platform’s dynamics. For example, because the platform’s mobility is contingent on sliding along its operating surface, it is highly sensitive to the irregularities present in that surface. Also, because of the low stiffness associated with PLA, the platform’s flexure mechanisms experience non-negligible transient behavior when given sawtooth inputs. These transients diminish the assumption of distinct stick and slip phases of motion as the driving frequency increases. Furthermore, the tether used to deliver electrical drive signals can be expected to contribute significantly to open-loop tracking error. As the legs simultaneously slip, the platform becomes highly susceptible to disturbances acting on its body. The tether introduces such a disturbance in the form of tension pulling the body in the direction along the tether. While care is taken to relieve this tension during testing, it is still expected to contribute heavily to the resulting tracking error. Finally, piezoelectric actuators, like those used in the platform’s legs, notoriously exhibit creep and hysteresis behaviors, both of which must be addressed for high precision applications. For these reasons, open-loop control of the platform’s positioning can not be expected result in repeatable, high-precision manipulation of samples.

While effective open-loop control of its motion may be infeasible, the microrobotic platform is to be used in imaging applications where visual data is implicitly available. By tracking features from either a sample or from the platform itself as it moves, the position of the platform can be visually servoed, leading to an ideal method for feedback control. Simulations of this closed-loop feedback approach are developed in Mathworks Simulink using the experimental results, observed trajectory deviations, kinematic models, and dynamic presented in the previous sections.

The measured frequency responses in Fig. 7 are used to determine a dynamic model of the individual flexure mechanisms’ displacements related to the input voltage, \(G_p(s)\). A second-order transfer function is assumed, as the first resonance of each leg is almost ten times the frequency of the given stick-slip trajectories of this work. Larger differences between the model and the physical system are expected at higher frequencies which would cause notable differences between the simulated and measured displacements. However, for the trajectories used in this paper the lower order model is assumed to be a sufficient approximation. The modeled transfer function is as follows:
$$\begin{aligned} G_p(s) = \frac{X(s)}{U(s)} = \frac{2\times 10^8}{s^2+1200s+2.956\times 10^7}, \end{aligned}$$
(11)
where s corresponds to the Laplace variable. The flexure’s velocity, \(G_v(s)\) is then the time derivative of this transfer function:
$$\begin{aligned} G_v(s) = \frac{(2\times 10^8) s}{s^2+1200s+2.956\times 10^7}. \end{aligned}$$
(12)
If the interactions between the surface and the flexure is neglected, the time response of an individual leg is found using the inverse Laplace transformation,
$$\begin{aligned} v(t) = \mathscr {L}^{-1}\{U(s)\cdot G_v(s)\}(t). \end{aligned}$$
(13)
To determine the resulting velocity of the body relative to the flexure leg, the leg’s contact with the surface is treated as either full-slip or full-stick. That is, if the leg velocity exceeds a specific value \(\rho\), the leg itself is moving, otherwise the robot body experiences the relative velocity in the opposite direction. Thus, the robot body velocity, \(v_b\) is determined to be
$$\begin{aligned} v_b(t) = \bigg \{\begin{array}{ll} -v(t) &{} \text{if }|v(t)|<\rho , \\ 0 &{} \text{otherwise}. \end{array} \end{aligned}$$
(14)
The body’s displacement, \(x_b\), is then the integral of its velocity:
$$\begin{aligned} x_b(t) = \int _0^{t} v_b(\tau )d\tau . \end{aligned}$$
(15)
The accumulative body displacement is expressed as an iterative calculation by only integrating \(v_b(t)\) over the cycle of interest. Thus the body position for step \(k+1\) given a trajectory time step of \({\varDelta } t\) is
$$\begin{aligned} x_b[k+1] = x_b[k] + \int _{k {\varDelta } t}^{(k+1) {\varDelta } t}v_b(\tau )d\tau . \end{aligned}$$
(16)
Fig. 9

Kinematic and dynamic leg models simulated at a 1 Hz and b 50 Hz. Top plots show total platform displacement and bottom plots detail the relative leg motion

While the above formulation assumes all three legs operate identically with the averaged, second-order dynamic model, it can be adjusted to higher-order individual leg models if more accurate simulations for higher frequency trajectories are desired. Figure 9 shows simulated results of the dynamic leg model for a single flexure, as well as the expected behavior for a purely kinematic response. At low frequencies the dynamic and kinematic leg models behave almost identically for the total body displacement, with some noticeable transient effects at the sawtooth input’s drop points. This is simulated at 1 Hz in Fig. 9a. At higher frequencies, the body’s net displacement begins to degrade for the dynamic model, while the kinematic response remains the same. This is simulated at 50 Hz in Fig. 9b. The leg response shows that more slipping occurs at the drop point and the resulting trajectory lags behind the kinematic model. At high enough frequencies, both models experience velocities that consistently exceed the threshold, \(\rho\). At these frequencies, no displacement is experienced as the mechanism is entirely in the slip phase.

The resulting model of the platform’s motion is augmented with the locomotive disturbances and directional coupling seen in the previous section. When provided with the same open-loop control law and trajectories as in experimentation, this full model results in the trajectory shown in Fig. 10 for stepping speed of 100 steps/s. This simulated tracking response exhibits orientation and translation errors similar to the empirical tracking response, proving it an acceptable open-loop model to demonstrate the benefits of closed-loop control.
Fig. 10

Simulated tracking of 10 mm square using open- and closed-loop control of the microrobotic platform. In all cases, stepping frequency of \(f_{step}=~100\) Hz is used. For each closed loop control case, camera sampling speed T is varied as it provides the feedback signal for a proportional control law with \(K_{x}=K_{y}=K_{\theta }=0.01\). The camera introduces Gaussian noise with standard deviation, \(\sigma = 1\) mm, which is mitigated by a unity low-pass filter with cutoff, \(f_{co}=10\) rad/s

Position data is gathered from a simulated imaging device with sampling time, T. The data is then corrupted using zero-mean Gaussian noise with standard deviation of \(\sigma = 1\) mm. A first-order, low-pass filter with cutoff frequency, \(f_{co}=10\) radians/s, is used to decrease the presence of this noise in the feedback signal, which is then incorporated into a proportional control law with constant gains of \(K_{x}=K_{y}=K_{\theta }=0.01\). These gains are used to mitigate errors in x-position, y-position, and angular orientation, \(\theta\) respectively.

In this simulated control scheme, the platform operates exclusively in stepping-mode, with constant stepping frequency, \(f_{step}\), and step sizes and directions modulated by the error signal and controller gains. If necessary, further precision could be achieved using a control scheme that also incorporates scanning-mode operation when the error signal is sufficiently small. One such controller could use the error signal to proportionally modulate stepping frequency, working to prevent the platform from entering the slip-phase when the error is sufficiently small. However, the level of noise introduced by the simulated sensor makes such precision unnecessary.

When applying the developed controller to tracking squares with side lengths of 10 mm at stepping frequency, \(f_{step} = 100\) Hz, the tracking precision increases inversely with the camera’s sampling time. This intuitively occurs because as sampling decreases, the camera captures the robot’s position in time and adjusts its trajectory more frequently, allowing it to adhere better to the desired path. The open-loop tracking behavior of the simulated model closely adheres to that of the experimental results presented earlier, with maximum tracking error (\(e_{max}\)) of 3.07 mm and RMS error (\(e_{rms}\)) of 1.9 mm. After introducing proportional feedback control, these RMS errors are reduced by as much as 90% for faster camera sampling times. Furthermore, \(e_{max}\) can be seen to increase with direct relation to the sampling time, demonstrating an expected degradation in tracking performance as position information becomes less frequent. Nonetheless, by rejecting both translational errors and induced orientation error while tracking, this simulation demonstrates the platform’s potential for 3-DOF trajectory tracking with high precision. The resulting tracking errors from these simulations are presented in Table 3. Also as expected, the induced errors from closed-loop control are of the same order as the noise introduced by the camera, meaning that the sensor limits precision in this application.
Table 3

Error induced by open- and closed-loop tracking of square trajectories

Control type

Sample time, T (s)

\(e_{max}\) (mm)

\(e_{RMS}\) (mm)

Open-loop (experimental)

NA

3.1

2.1

Open-loop (simulated)

NA

3.07

1.9

Closed-loop (simulated)

0.2

0.353

0.214

Closed-loop (simulated)

1.0

0.419

0.194

Closed-loop (simulated)

2.5

0.650

0.302

3.5 Experimental closed-loop control via visual servoing

Closed-loop control is implemented using a similar visual servoing scheme introduced in the prior section. The same prototype platform and electromechanical hardware introduced previously are used in addition to a digital microscope (Dino-Lite AM3111T), which provides visual feedback.

In creating a practical visual control algorithm, feature points could be extracted from a particular sample and used to servo that sample’s position and orientation. For demonstrative purposes however, a glyph comprised of a solid black 1 cm \(\times\) 0.25 cm rectangle is mounted to the platform, allowing for simplified feature extraction from recorded images. Feature extraction is performed by first thresholding a recorded gray-scale image from the microscope. An area filter is then used to isolate the glyph, from which its centroid and major axis orientation are computed. The platform’s position and orientation are determined from these features and compared to desired trajectories from a trajectory planner, resulting in an error signal that can be penalized by the controller.

As with the simulated closed-loop control, a proportional controller with three gain terms, \(K_{x}\), \(K_{y}\), and \(K_{\theta }\), are used to penalize errors in the platform’s x-position, y-position, and angular orientation, respectively. These gains are tuned to \(K_{x} = 2\), \(K_{y} = 2\), and \(K_{\theta } = 20\) after empirical analyses of tracking response data.

The trajectory planning stage is set to navigate the platform through a square trajectory with side length of 5 mm. During this testing, one central processor (Intel Core i5-5675R) utilizes MATLAB development tools to generate the platform’s trajectories, run image processing algorithms, and compute control signals that are then sent to the data acquisition device through an RS-232 protocol. As a result of attempting to run these processes simultaneously, the prototype platform can only reliably achieve a stepping speed of 5 Hz while visual data is being recorded and processed. Therefore, to allow for sufficient time to compensate for error while tracking the square trajectory, each side is traversed over the course of 6000 s, with the resulting tracking behavior shown in Fig. 11.

To mitigate the processing constraints associated with control, electrical drivers and microcontrollers can be placed on-board the platform, allowing for the computing load to be distributed between the platform and a host system as well as software to be optimized for both systems.

As seen with the simulations performed in the preceding section, the maximum and RMS errors induced while tracking are again limited by the noise seen in the recorded images. In this case, the noise seen from the digital microscope being used is on the order of 0.05 mm. This, combined with a camera frame rate of 10 frames/s, results in significantly increased tracking performance relative to the simulated results in the prior section. While tracking the square, position and orientation errors remain under 0.095 mm and 0.49\(^{\circ }\), respectively, and the platform exhibits RMS errors of 0.024 mm and 0.132\(^{\circ }\)—a 98.9% improvement in comparison to the open-loop square-trajectory-tracking presented previously.
Fig. 11

Experimental tracking of a 5 mm square implementing closed-loop control via visual servoing of the microrobotic platform. A stepping speed of 5 Hz is used along with proportional control gains of \(K_{p_{x}} = 2\), \(K_{p_{y}} = 2\), and \(K_{p_{\theta }} = 20\). The digital microscope used for visual feedback introduces noise on the order of 0.05 mm, which proves to limit the platform’s tracking performance

4 Conclusions

The design, prototyping, and control of a novel 3D-printed, 3-DOF microrobot with sub-micrometer precision and unrestricted travel range was presented in this article. Piezoelectric stick-slip actuators augmented with mechanical amplifying mechanisms to improve the stroke lengths and resonant behaviors are implemented as the actuation mechanism, and the arrangement of these PSS actuators was used to determine a forward kinematic model predicting the total microrobot translation and rotation according to individual actuator inputs over a single stick-slip cycle. This model was then inverted for feedforward control of the robot.

A prototype platform incorporating the PSS tripedal architecture was manufactured and characterized. Individual PSS actuators experienced maximum stroke lengths of approximately 29.4 \(\upmu\)m, with first resonances occurring, on average, at approximately 860 Hz. The kinematic model was used for trajectory tracking on the experimental system, demonstrating a Euclidean RMS error of 2.1 mm for a 3 mm \(\times\) 3 mm square. Feedback control was simulated using visual servoing of the platform with proportional error compensation, resulting in a decrease in tracking error by almost 90%. The simulated control scheme was then applied to the prototype platform, resulting in a 99% decrease in tracking error relative to open-loop control.

In future work, drive electronics will be embedded on-board the platform, removing the necessity and constraining effects of the attached electrical tether. Furthermore, the models and control schemes discussed in this paper will be augmented and adapted with more advanced algorithms to improve tracking performance in both open- and closed-loop cases. For example, calibration routines will be developed to compensate for directional coupling and frictional effects of a given surface. This improvement is expected to significantly increase open-loop tracking performance.

Notes

Acknowledgements

This material is based upon work supported, in part, by the National Science Foundation Grant nos. CMMI 1537983, 1537722, and 1708536. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Copyright information

© Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  1. 1.Department of Mechanical Engineering, University of Utah Robotics CenterUniversity of UtahSalt Lake CityUSA

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