# Arm-Z as a Modular Tracking Device

Ela Zawidzka, Jacek Szklarski, Machi Zawidzki

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Ela Zawidzka, Jacek Szklarski, Machi Zawidzki. Arm-Z as a Modular Tracking Device. Proceedings of Seventh International Congress on Information and Communication Technology, Feb 2022, London, Poland. pp.429-437, 10.1007/978-981-19-1610-6_37. hal-04969396

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# Arm-Z as a modular tracking device

Ela Zawidzka, Jacek Szklarski, and Machi Zawidzki

Institute of Fundamental Technological Research, Polish Academy of Sciences,
zawidzki@ippt.pan.pl

**Abstract.** Arm-Z is a concept of a robotic manipulator comprised of linearly joined congruent modules with possibility of relative twist. The advantages of Arm-Z are: economization (mass-production) and robustness (modules which failed can be replaced, also if some fail the system can perform certain tasks). Non-intuitive and difficult control are the disadvantages of Arm-Z. This paper presents a concept of a modular tracking device comprised of four congruent modules. As an example the Sun-tracking setup is used with possible application for solar energy harvesting.

**Keywords:** Extremely Modular System, Arm-Z, Hyper-redundant manipulator, Sun-tracking

## 1 Introduction

The snake mechanism is a redundant system which, however, makes snakes supremely adapted for various habitats. Analogously, in irregular environments, bio-inspired trunk-like robots in some cases perform better than more conventional wheeled, tracked and legged forms of robotic mobility. Research on snake robots has been conducted for several decades. Snake locomotion has been studied empirically already in 1940s [1]. 50 years later, the first mathematical model has been developed and trunk-like locomotors and manipulators have been proposed in [2]. Trunk-like manipulators have very specific movement characteristics which makes them especially useful in situations where classic robotic arms cannot be used. Such manipulators can transport working heads in very complex, narrow, confined, cluttered spaces which are inaccessible by any other means. Reaching range can be of order of tens of meters and actions can be accomplished in hazardous environments. They can perform tasks like welding, cleaning, visual inspection and any other which can be realized by the head mounted on the manipulator tips. It should be stressed that the existing solutions in this field are very limited. Only a few companies offer such type of manipulators, and these are rather experimental (Fig. 1).
<figure><img src="image_2.png" /><figcaption>Fig. 1: Examples of existing trunk-like robotic manipulators. OC Robotics arms (http://www.ocrobotics.com).</figcaption></figure>

Additionally, trunk/snake-like manipulators may have relatively large num-
ber of degrees of freedom (DOF) (however, there are exceptions). This contrasts
with the classic robotic arms with low number of DOFs which are widely used
in the industry. In the case of Arm-Z, the number of DOF is equal to the total
number of the connected modules. Besides the possibility of performing complex
actions, large number of DOF introduces the possibility of high fault tolerance
and robustness. It aligns well with the concept of hyper-redundant manipulators
(HRM, [3]). Their advantages come at a price of notoriously difficult control due
to their highly non-linear nature. E.g. the inverse kinematics problem for a typi-
cal industrial robotic arm can be solved relatively easily [4] and thus the control
of such arms is straightforward. On the other hand, the control of a *bionic trunk*
requires sophisticated artificial intelligence methods [5–7]. However, in return,
one can profit from all the important advantages of trunk-like HRMs. For further
details on this class of robotic manipulators see [8].

The concept of extreme modularization is relatively new, and less explored.
In general, research on structural optimization of modular structures seems to
be relatively sparse. Ref. [9] considers layout optimization of a modular bridge,
where the design variables are effectively discrete and encode the choice from a
predefined set of local module topologies. Simultaneous optimization of a family
of a few vehicle structures at the level of the shared basic beam components is
discussed in [10].

## 2 The concept of Arm-Z

Complex three-dimensional tubular forms can be built with relatively simple
congruent units as presented in [11]. A parametric design system comprised of
congruent modules was introduced in [12].

Arm-Z is a conceptual manipulator composed of congruent modules each
having one degree of freedom (1-DOF) - a relative twist ([13]). Simple changes
of these twists result in emerging behavior of the entire Arm-Z allowing it to
perform complex movements. The relative twists can be continuous or discrete.
Arm-Z modules are geometrical objects analog to a sector of circular torus. Each
module is defined by parameters: number of sides $k$, size $r$, offset $d$, and angle $\zeta$ between bottom (B) and top (T) surfaces. Fig. 2 illustrates these parameters.

<figure><img src="image_3.png" /><figcaption>Fig. 2: On the left: visualization of basic hexagonal Arm-Z module defined by parameters: $r$, $d$, $\zeta$ and $k = 6$ and its axonometric view. On the right: examples of simple assembly of modules with various combinations of their geometrical parameters.</figcaption></figure>

The overall shape of Arm-Z is a result of: the geometrical parameters of each module, number of modules and their relative rotation. Fig. 3 shows two Arm-Z modules at six consecutive dihedral twists from 0 to $\pi$.

<figure><img src="image_4.png" /><figcaption>Fig. 3: The relative twist of Arm-Z modules. The first (bottom) module is fixed and indicated in gray. The axis of rotation for the subsequent module is shown in red.</figcaption></figure>

The operational space depends on the number of modules. It forms a complex 3D surface which has voids. Therefore, in general, the tip of Arm-Z can not reach all points in 3D space within its range, but can reach specific points with certain accuracy (error), as illustrated with volumetric renderings in Fig. 4.
<figure><img src="image_5.png" /><figcaption>Fig. 4: Volumetric renderings showing surfaces of reaching error for various N<sub>dof</sub>. It is clear that the largest errors are close the base of the Arm-Z structure and in the corners of the bounding box.</figcaption></figure>

## 3 The Sun-tracking task

As an example of a steady movement of the manipulator following given direction we propose a sun-tracking device with possible application e.g. for solar energy harvesting. Fig. 5 shows the position of the Arm-Z in the coordinate system and explains the tracking parameters. **V_N** is the normal vector to the surface of the last module, in other words, it is a vector pointing from the tip of the manipulator. **V_d** stands for the desired direction of the normal vector (in this case, the sunlight direction). **V_d**(*φ*, *θ*) is defined by the azimuth *φ* and altitude *θ*. In this task, **V_N** must follow **V_d**, in other words the angle between these two vectors is minimized. **P_t** is the position of the tip of the manipulator. *N*<sub>dof</sub> is the number of degrees of freedom of the system, it is equal to the number of modules - 1.
<figure><img src="image_6.png" /><figcaption>Fig. 5: The coordinate system, notation and the control parameters.</figcaption></figure>

In this task the intention is to follow the Sun direction during summertime (from summer solstice to September equinox) between 10 a.m. and 6 p.m., as illustrated in Fig. 6.

<figure><img src="image_7.png" /><figcaption>Fig. 6: Possible sun location during a year and for the given periods during summertime (indicated in red).</figcaption></figure>
The angular errors for manipulators of various numbers of degrees of freedom: 1,2,3,4,6 and 7 are shown in Fig. 7. White line indicates there is no horizontal plane. Black indicates very small errors. Already a 4-module manipulator can cover almost all desired directions. The last figure shows that 7 degrees of freedom allow to cover almost all directions in 3D space (including the space below the manipulator).

<figure><img src="image_8.png" /><figcaption>Fig. 7: Visualization of the angular errors for Arm-Z manipulators of various numbers of degrees of freedom: 1,2,3,4,6 and 7. The outline indicating the tracking period from Fig. 6 is superimposed as the white dashed line for the case of 3 DOF.</figcaption></figure>
## 3.1 The optimization of the tracking action

In order to perform realistic simulations, the manipulator has been implemented in Bullet Physics SDK [14]. It is a widely used tool for real-time collision detection and multi-physics simulation for virtual reality (VR), robotics, machine learning etc. In order to control the manipulator, the inverse dynamics has been solved by two numerical methods: the classical recursive Newton Euler algorithm (RNEA), [15] and by means of dual annealing [16, 17]. As it turned out, the latter method performed better (in terms of average errors) than RNEA. This can be attributed to the fact, that the system is highly nonlinear and even small changes in position of modules at the bottom can profoundly influence the tip position.

In order to continuously control the manipulator, the twist velocities and spins of three modules have been optimized with the dual annealing method in a way in which significant deviations from the current state have been penalized. This made possible to achieve a quite smooth movement of the manipulator. The results (angular velocities of the individual modules) are shown in Fig. 8.

<figure><img src="image_9.png" /><figcaption>Fig. 8: Joint states $x_1, x_2, x_3$ for a four-module structure which follows the sun on July 1st between 8 a.m. and 8 p.m. local time.</figcaption></figure>

Fig. 9 shows the angular errors throughout the tracking period.
<figure><img src="image_10.png" /><figcaption>Fig. 9: Error (angle between $V_d$ and $V_N$) for globally optimal states and for the continuous control scenario (some peaks can be noticed for the latter). From Fig. 7, it follows that the tips normal vector $V_N$ can not follow the sun for low latitudes, especially during sunrise/sunset. Hence the growing error from approximately 6 p.m.</figcaption></figure>

## 4 Conclusions

- Effective direction-tracking in given task is already possible with only four modules (of which three are moving, thus 3 DOF).

- The precision of direction-tracking in this simple example is satisfactory.

## 5 Future Work

- Improvement of the control

- Development of larger Arm-Z for more demanding tasks

- Building a prototype

## Acknowledgements

This research is a part of the project titled *Arm-Z: an extremely modular hyper-redundant low-cost manipulator – development of control methods and efficiency analysis* and funded by OPUS 17 research grant No. 2019/33/B/ST8/02791 supported by the National Science Center, Poland.
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