## 4.6. Comparisons

In this part, comparisons among the proposed control scheme and existing methods are given to show the superiority of the RNN based strategy. The comparisons are shown in **Table 1**. In Guo and Zhang (2012), a RNN based controller is designed for redundant manipulators, both obstacle avoidance and physical constraints are considered. However, the controller only focus on kinematic control problem. In Nanayakkara et al. (2001) and Csiszar et al. (2011), force control together with obstacle avoidance are taken into account, but the physical constraints are ignored. Xu et al. (2019a) develop an adaptive admittance control strategy, which is capable of dealing with force control under model uncertainties, physical constraints and real-time optimization. It is remarkable that the proposed strategy focus on real-time obstacle avoidance in force control tasks, and the controller is capable of ensuring the boundedness of joint angles and velocities. At the same time, simulations have shown the potential of optimization ability of norm of joint speed.

## 5. CONCLUSIONS

In this paper, a novel collision-free compliance controller is constructed based on the idea of QP programming and neural networks. Different with existing methods, in this paper, the control problem is described from an optimization perspective, and the compliance control and collision avoidance are formulated as equality or inequality constraints. The physical constraints such as limitations of joint angles and velocities are also taken into consideration. Before ending this paper, it is worth pointing out that it is the first RNN based compliance control method, which considers collision avoidance problem in realtime, and also shows great potential in handling physical limitations. In this paper, simple numerical simulations in MATLAB are carried out to verify the efficiency of the proposed controller. In the future, we will check the control framework with different impedance models in physically realistic simulation environments, and then consider machine vision technology and system delay problem on physical experimental platforms.

FIGURE 9 | Simulation results of the established RNN in a narrow workspace. (A) Is the profile of λ1. (B) Is the profile of λ2. (C) Is the profiles of the desired and reference trajectory along x-axis. (D) Is the profiles of the desired and reference trajectory along y-axis. (E) Is the profiles of the objective function of the proposed controller and JPMI based method.

## DATA AVAILABILITY

All datasets analyzed for this study are included in the manuscript and the supplementary files.

## AUTHOR CONTRIBUTIONS

XZ designed the control strategy and writes the paper. ZX did the numerical experiments. SL revised the paper and proposed amendments.

## FUNDING

This work was supported by the GDAS' Foundation of National and Provincial Science and Technology Talent (Grant No. 2018GDASCX-0603), Guangdong Province Applied Science and Technology Research Project (Grant No. 2015B090922010), Guangdong Province Public Welfare Research and Capacity Building Project (Grant No. 2017A010102017), Guangdong Province Science and Technology Major Projects (Grant No. 2016B090910003), Guangdong Province Science and Technology Major Projects (Grant No. 2017B010116005), Guangdong Province Key Areas R&D Program (Grant No. 2019B090919002).

## ACKNOWLEDGMENTS

The authors would like to thank the reviewers for their valuable comments and suggestions.

## REFERENCES

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Zhou, Xu and Li. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Series Elastic Behavior of Biarticular Muscle-Tendon Structure in a Robotic Leg

Felix Ruppert\* and Alexander Badri-Spröwitz

Dynamic Locomotion Group, Max Planck Institute for Intelligent Systems, Stuttgart, Germany

We investigate the role of lower leg muscle-tendon structures in providing serial elastic behavior to the hip actuator. We present a leg design with physical elastic elements in leg angle and virtual leg axis direction, and its impact onto energy efficient legged locomotion. By testing and comparing two robotic lower leg spring configurations, we can provide potential explanations of the functionality of similar animal leg morphologies with lower leg muscle-tendon network structures. We investigate the effects of leg angle compliance during locomotion. In a proof of concept, we show that a leg with a gastrocnemius inspired elasticity possesses elastic components that deflect in leg angle directions. The leg design with elastic elements in leg angle direction can store hip actuator energy in the series elastic element. We then show the leg's advantages in mechanical design in a vertical drop experiment. In the drop experiments the biarticular leg requires 46% less power. During drop loading, the leg adapts its posture and stores the energy in its springs. The increased energy storing capacity in leg angle direction reduces energy requirements and cost of transport by 31% during dynamic hopping to a cost of transport of 1.2 at 0.9 kg body weight. The biarticular robot leg design has major advantages, especially compared to more traditional robot designs. Despite its high degree of under-actuation, it is easy to converge into and maintain dynamic hopping locomotion. The presented control is based on a simple-to-implement, feed-forward pattern generator. The biarticular legs lightweight design can be rapidly assembled and is largely made from elements created by rapid prototyping. At the same time it is robust, and passively withstands drops from 200% body height. The biarticular leg shows, to the best of the authors' knowledge, the lowest achieved relative cost of transport documented for all dynamically hopping and running robots of 64% of a comparable natural runner's COT.

Keywords: robotics, bioinspired robotics, intrinsic compliance, locomotion, energy efficiency, biarticular, leg design, series elastics

## 1. INTRODUCTION

A persistent question in legged locomotion relates to the functional morphology of compliant elements in segmented leg structures. Elastic elements in legs enhance locomotion performance in terms of stability, robustness to perturbations and impact mitigation in legged walking systems (Hurst, 2008; Rummel et al., 2010). Leg elasticity can simplify the control task (Verstraten et al., 2016; Beckerle et al., 2017) by giving the system favorable natural dynamics. Biological observations

#### Edited by: Zhao Guo,

Wuhan University, China

Reviewed by: Muye Pang, Wuhan University of Technology, China Ning Tan, Sun Yat-sen University, China

> \*Correspondence: Felix Ruppert<EMAIL_ADDRESS>
Received: 18 April 2019 Accepted: 22 July 2019 Published: 13 August 2019

#### Citation:

Ruppert F and Badri-Spröwitz A (2019) Series Elastic Behavior of Biarticular Muscle-Tendon Structure in a Robotic Leg. Front. Neurorobot. 13:64. doi: 10.3389/fnbot.2019.00064 show that muscles and tendons act like elastic elements (Biewener, 1998; Alexander, 2002) that enable rich locomotion skills with high energy efficiency at low control effort (Daley, 2008; Lakatos et al., 2018).

In bioinspired robotics, the concept of elasticity was first introduced in series elastic actuators (SEA) (Pratt and Williamson, 1995) and prismatic actuators (Raibert et al., 1984). Many robotic designs use a minimal order template, the spring-loaded inverted pendulum (SLIP) model (Blickhan, 1989; Seyfarth et al., 2001; Geyer et al., 2006), as a design baseline for walking systems. Based on SLIP models much effort has gone into designing compliance in virtual leg axis direction for robots. Compliance is implemented as either motor controlled compliance (Ding and Park, 2017; Park et al., 2017) or physical springs (Fukuoka et al., 2003; Renjewski et al., 2015; Semini et al., 2015). Like the SLIP model, these robots have elastic elements in their joints to help them achieve the same energy efficient and robust behavior as their biological role models (Alexander and Bennet-Clark, 1977).

The primary focus in designing compliance in robots using physical springs has been on virtual leg axis direction compliance. We pronounce the influence of additional physical elastic elements acting in leg angle direction. In a real world locomotion scenario we show compliance in leg angle direction to be important as well as compliance in virtual leg axis direction which has been shown in SLIP model and SLIP-inspired robots. To achieve intrinsic compliance we implement a mechanism inspired by a biological blueprint.

In studies of quadrupedal leg morphology, a four-barlike mechanism has been observed by Lombard (1903). This simplified mechanism describes the functional morphology of lower leg muscle-tendon groups. It was extended by Witte et al. (2001, 2004) to a pantograph structure, including muscle-tendon structures. Because of the distal elastic tendon structures (Roberts, 2016), the simplified pantograph structure is spring-loaded. The concept implemented in a robot (Spröwitz et al., 2013), briefly suggested a potential function as an effective elastic element in leg angle direction (Spröwitz et al., 2014). The element is oriented so that its elastic elements possess deflection components orthogonal to virtual leg axis direction. Unlike in SLIP model, these components do not primarily contribute to deflection in virtual leg axis direction. However, they deflect under the presence of hip torque and perturbations that reflect as a torque to the hip actuator.

This leg morphology has been applied in robots before, empirically showing its advantages concerning the simplification in creating stable gaits. However, the general morphology has not yet been characterized, and the differences and advantages are not yet documented.

We investigate the effects, leg angle compliance in combination with virtual leg axis compliance has on spring behavior and resulting energy efficiency in the leg.

In this paper, we characterize one leg design with virtual leg axis compliance and one with virtual leg axis and leg angle compliance. We show the differences in leg morphology first on a simple kinematic model. To decompose virtual leg axis and leg angle effects we conduct static experiments to examine isolated virtual leg axis and torque influence on the elastic elements. In a drop test experiment, we investigate the mechanical behavior under dynamic loading without considering control design. At last, we compare both legs in a monoped hopping experiment and analyze the differences in dynamic behavior and energy stored and recuperated in the springs under a realistic load case.

## 1.1. Related Work

The functional morphology of multiple degrees of compliance in multi-segmented legs in animals and robotics has not been understood yet by either biologists nor roboticists. While twosegmented legs with one degree of compliance have been studied thoroughly (Raibert et al., 1984; Hutter et al., 2012; Semini et al., 2015; Park et al., 2017), the placement and interplay between multiple compliant elements is still an unsolved research topic.

Because of observations in biological examples, implementations of multi-segmented legs with several compliant elements have been tested in robotic hardware as well as in simulations to understand their behavior. Spröwitz et al. (2013, 2018) implemented a leg with a biarticular spring to investigate self stabilizing behavior on a quadruped during dynamic locomotion. They showed, that a simple sensorless central pattern generator with a position controller can allow dynamic feed forward locomotion. Iida et al. (2007) investigated the possibility to create both walking and running gaits in a humanoid biped with biarticular springs as well as the ability to create more human-like gaits. Sato et al. (2015) implemented a robot with only one biarticular spring but no intrinsic compliant knee. There, the biarticular spring provided elastic behavior to the leg for jumping and landing motions.

An aspect that has not been in the research focus yet is the interplay between both an intrinsically compliant knee and a biarticular spring in a multi-segmented leg. No systematic and comparative research exists so far comparing multiple compliant elements in highly under-actuated segmented legs, specifically for the combination of leg-angle and virtual leg axis compliance. As energy fluctuates in both directions in animal legs (Alexander, 1984) one can expect that compliant passive mechanisms evolved benefiting from these resources, i.e. energetically. We focus our research on the torque influence onto a series elastic biarticular spring and the increase in energy efficiency the additional stored energy provides.

In this paper we present a leg design with compliance in virtual leg axis direction as well as in leg angle direction. We show that the element in leg angle direction charges under torque influence, providing series elastic behavior for the hip. We show how the implementation of this element can drastically increase the amount of elastic energy stored in the leg.

## 2. MATERIALS AND METHODS

## 2.1. Leg Design and Implementation

The bio-inspired leg designs under investigation (**Figure 1**) consist of three segments. A femur segment, a shank segment with a four-bar structure and a foot segment. The arrangement of segments and elastic elements (**Figure 2B**), is inspired by the

with angle definitions and symbol annotations.

via the patella. Gastrocnemius originates on the lower femur and inserts into the upper foot segment spanning the knee and ankle joint. (B) Photo of the biarticular leg mounted on the boom structure. The biarticular spring is hidden under the two parts of the biarticular segment; the point of contact is visible as a slit.

leg anatomy of mammalian quadrupeds. The hip joint connects the femur segment to the trunk; the knee joint connects the hip and shank segment; the ankle joint connects the shank and foot segment. Leg segments on these legs represent the major bone groups of vertebrate animals, namely femur, tibia and fibula and the bones forming the foot segment. For simplicity, all segments in both designs have the same length.

Elastic elements in the robot are placed to mimic the functionality of big muscle-tendon groups in animal legs. In placement and functionality, the knee spring represents the quadriceps femoris muscle and patella tendon of the biological example. On the pantograph leg the segment parallel to the shank segment of the biarticular leg consists of a rigid element, forming a pantograph structure (**Figure 1A**). The segment parallel to the shank segment consists of a second spring connecting the hip and foot segment (**Figure 1B**). This biarticular segment spans two joints. The biarticular spring models the lower leg muscletendon apparatus of gastrocnemius muscle and Achilles tendon in a quadrupedal animal (**Figure 2A**). We refer to the leg with the biarticular spring as the biarticular leg, to the leg with the pantograph structure as the pantograph leg. Unloaded, the biarticular segment has the same length as the shank segment. The femur and foot segment are parallel when the biarticular spring is not deflected. Both ankle and knee joint have a hard stop to prevent over-extension.

The knee joint stiffness is realized by a spring that wraps around the knee joint on a cam mechanism, inspired by a knee cap (patella) (Allen et al., 2017; Heim et al., 2018). The knee cam mechanism linearizes the knee spring deflection over knee angle. Knee stiffness is designed to provide sufficient torque to hold the leg during running, exerting ground reaction forces three times the body weight of the robot at 10% virtual leg length deflection. The cam radius on the knee is designed to enable 35◦ knee angle deflection or about 70 mm leg length change. Empirically, we choose the biarticular spring stiffness similar to the knee spring stiffness, so the biarticular spring does not saturate, and the knee spring deflects similar to the pantograph leg. Through the biarticular spring the ankle joint can deflect by 60◦ , equivalent to 160 mm leg length change. The hip joint is articulated with a brushless motor. In combination with a 5:1 planetary gear box the nominal output hip torque is 6.2 Nm. To measure the joint deflections, all joints on the leg are instrumented with rotary absolute encoders.

The leg design consists of a hip joint rigidly connected to an actuator and two passive joints on the knee and ankle. The leg design builds on previous research on the Cheetah cub and Oncilla robots. Instead of the servo motors used in the previous robots we implemented high torque density brushless motors. To increase the backdriveability of the gear train a low ratio gearbox was used. This way the actuator can potentially be used as a proprioceptive actuator (Seok et al., 2012).The new knee spring placement in our design largely reduces the nonlinearity of the spring force to joint angle relationship of the knee joint, compared to the design used in Cheetah-cub and Oncilla. This simplifies modeling, and reduces the complexity of the mechanical design. The general mechanical leg design was improved to be more dureable and robust while at the same time reducing the complexity of the design to enable faster prototyping, as well as simplified manufacturing and assembly. The biarticular leg's lightweight design can be rapidly assembled, and is largely made from elements created by rapid prototyping. At the same time it is however robust, and passively withstands drops from 200% body height.

Here we reduce the investigation to a single leg hopping in the saggital plane. This is common practice (Semini et al., 2008; Hutter et al., 2011; Ding and Park, 2017; Liu et al., 2018). It also reduces the effects of body inertia, multiple legs and the system complexity.

## 2.2. Kinematic Model

In this section, we investigate the governing equations describing the difference in behavior for both legs. All future assertions talk about joint angles implying resultant deflection of the associated elastic elements.

By formulating the kinematic equations for the pantograph leg, we show the basics of our hypothesis. Writing the forward kinematics to obtain the foot position with the reference at the hip joint, shows that the system rank r = 2 with 2 parameters (θhip and θknee), since θhip = θankle because of the pantograph structure. The equation system is fully defined. In comparison, the pantograph segment in the biarticular leg is replaced by a biarticular spring. The rank of the system matrix is also r = 2 but because θhip 6= θankle, an additional parameter or Degree Of Freedom (DoF) is added to the leg. Annotations are depicted in **Figure 1** and **Table 1**.

Forward kinematics for biarticular leg:

$$\begin{aligned} x\_{\text{foot}} &= -l \cdot \sin(\theta\_{\text{hip}}) + l \cdot \sin(\theta\_{\text{knee}}^{\text{g}}) - l \cdot \sin(\theta\_{\text{ankle}}^{\text{g}}) \\ y\_{\text{foot}} &= -l \cdot \cos(\theta\_{\text{hip}}) - l \cdot \cos(\theta\_{\text{knee}}^{\text{g}}) - l \cdot \cos(\theta\_{\text{ankle}}^{\text{g}}) \end{aligned} \quad \text{(2)}$$

with,

$$\begin{aligned} \theta\_{kne\varepsilon}^{\mathcal{g}} &= \pi - \theta\_{hip} - \theta\_{kne\varepsilon} \\ \text{and} \\ \theta\_{ankle}^{\mathcal{g}} &= \theta\_{kne\varepsilon}^{\mathcal{g}} - \theta\_{ankle} \end{aligned} \tag{3}$$

To describe the joint positions of the biarticular leg, an additional kinetic constraint is necessary to describe the coupling of the two springs:

$$\begin{aligned} \frac{r\_k \cdot F\_{knee} + \overrightarrow{r\_{pk}} \times \overrightarrow{F\_{biart}} + \tau\_{hip}}{\cos(\theta\_{hip}^{\mathcal{S}})} &= \\ \frac{F\_{\times} \cdot l \cdot \cos(\theta\_{ankle}^{\mathcal{S}}) + \overrightarrow{r\_{pa}} \times \overrightarrow{F\_{biart}}}{\cos(\theta\_{ankle}^{\mathcal{S}})} \end{aligned} \tag{4}$$

$$\begin{aligned} F\_{knee} &= k\_k \cdot r\_k \cdot \Delta\theta\_{knee} \\ \overrightarrow{F\_{biart}} &= k\_p \cdot \left( \overrightarrow{P} - l \cdot \frac{\overrightarrow{P}}{||\overrightarrow{P}||} \right) \end{aligned} \tag{5}$$

where, Fknee is the force of the knee spring and Fbiart is the force of the biarticular spring.

$$\begin{aligned} \overrightarrow{r\_{pa}} &= \begin{bmatrix} -||r\_{pk}|| \cdot \sin(\theta\_{ankle}^{\mathcal{S}}) \\ ||r\_{pk}|| \cdot \cos(\theta\_{ankle}^{\mathcal{S}}) \end{bmatrix} \\ \text{and} \\ \begin{aligned} \end{aligned} \quad \text{and} \\ \begin{aligned} \begin{array}{l} \blacksquare & ||r\_{pk}|| \cdot \sin(\theta\_{ik<}) - l\_{\cdot} \sin(\theta\_{i}^{\mathcal{S}}) - ||r\_{\cdot}|| \cdot \sin(\theta\_{\cdot}^{\mathcal{S}}) \end{array} \end{aligned} \quad \text{and} \\ \begin{aligned} \begin{array}{l} \blacksquare & \square & \square & \square & \square \end{array} \end{aligned} $$

$$\overrightarrow{P} = \begin{bmatrix} ||r\_{pk}|| \cdot \sin(\theta\_{hip}) - l \cdot \sin(\theta\_{knee}^{\p}) - ||r\_{pa}|| \cdot \sin(\theta\_{ankle}^{\p}) \\ - ||r\_{pk}|| \cdot \cos(\theta\_{hip}) - l \cdot \cos(\theta\_{knee}^{\p}) + ||r\_{pa}|| \cdot \cos(\theta\_{ankle}^{\p}) \end{bmatrix} \tag{6}$$

where, −→<sup>P</sup> is the position of the biarticular spring insertion into the foot segment. If we assume the hip and foot fixed with rotary joints to a global frame (**Figure 3**), the pantograph leg cannot change its joint angles. Because of the increased DOF, the biarticular leg has an infinite number of joint orientations with a fixed hip and foot point.

By changing the torques and forces acting on the biarticular leg, the joint orientation can be changed based on the ratio of chosen stiffnesses. Under hip torque, the pantograph leg increases the forces on the hip and foot bearings but does not change joint angles. The biarticular leg orients its joints to satisfy the kinetic constraint described above. By changing its posture, the biarticular leg deflects the springs attached to each joint. When torque is exerted on the hip, the leg can store the energy from hip actuation in the biarticular spring. The energy storage potentially enables the biarticular leg to recuperate the energy stored in the springs.

Spring energies for leg comparison are calculated as:

$$\begin{aligned} E\_{kne\varepsilon} &= \frac{(F\_{kne\varepsilon})^2}{2 \cdot k\_k} \\ &\text{and} \\ E\_{biant} &= \frac{(F\_{birrt})^2}{2 \cdot k\_{birrt}} \end{aligned} \tag{7}$$

where E<sup>i</sup> is the energy stored in the corresponding spring, kk is the knee spring stiffness, and kbiart is the biarticular spring stiffness.

## 2.3. Experiments

During locomotion, legs are subject to dynamic forces in leg length, as well as leg angle direction. In this section, we investigate the behavior of both legs under loads in both directions. To show the basic functionality of the leg we reduce the experiment complexity compared to a hopping experiment. In a reduced order experiment, we investigate the effects of virtual leg axis and leg angle forces separately. Then we investigate the mechanical leg behavior in a vertical drop test without control influence. Last we show that the leg shows series elastic behavior in the biarticular spring under combined loads during dynamic hopping to provide a realistic locomotion load case.

## 2.3.1. Static Virtual Leg Axis Forces

First, we implemented a simplified setup neglecting weight and inertia effects to show the virtual leg axis related behavior clearly. Both the foot and hip joint of each leg were fixed to a ground frame by a rotational pin joint. The joint restricts both hinge points to one rotational DOF (**Figure 3A**). The ball bearings used to implement the rotational joints only allow forces to be transmitted, but no torques. This experiment investigates the change in joint angles purely based on change in virtual leg length. We fixed the hip joint to the frame at different virtual leg lengths in steps of 5 mm from resting leg length to 65 mm deflection, and measure the joint angles with rotary encoders.

## 2.3.2. Static Leg Angle Torque

In the next step, we investigated the effects of hip torques on the legs in the static test stand and observed the joint angles. The legs were fixed to the same static test setup as before. Both legs were deflected by 10 mm initial leg length. We applied hip torque from 0 to 2.5 Nm in steps of 0.1 Nm every 2 s to exclude acceleration effects. We measured the resulting joint deflections as well as the forces exerted onto the foot fixture with a force sensor (K3D60 me-systeme) to verify the applied hip torque.

## 2.3.3. Vertical Drop Experiment

After investigating the static behavior of the leg, we focus on behavior under dynamic loading. We separate the effects of virtual leg axis forces and leg angle torque for vaulting the leg during forward hopping with a vertical drop experiment (**Figure 4**). Holding the legs at a defined position requires a motor with a position controller. We want to investigate only the mechanical response to dynamic virtual leg axis forces without effects induced by the controller. Using the same controller could

give advantages to one leg, and different controllers for both legs would be hard to compare in mechanical performance. To eliminate this potential bias, we implemented a virtual spring on the hip actuator. The motor mimicked a torsion spring between the hip joint and a global frame. The virtual spring had its set point at 17.5◦ . At this hip angle, the virtual leg was vertical. By using a direct drive motor as a proprioceptive actuator, we avoided measuring inaccuracies through backlash, friction and reflected gearbox inertia. We used the proprioceptive actuator as a sensor to directly measure the hip angle deflection as well as the resulting forces. The virtual spring stiffness was chosen at 5.8 Nm rad to match the position controller gains used for the hopping experiments. The leg with joint encoders was connected to a boom structure, to restrict the motion to the sagittal plane (**Figure 2B**). The boom was instrumented with rotational encoders (AMT 102-V) to measure the horizontal and vertical angle of the boom representing the center of mass

pantograph segment left. Spring loaded biarticular segment right.

FIGURE 4 | High-speed snapshots of drop experiment starting from release to next hip apex. Pantograph leg top row, biarticular leg bottom row. Depicted are left to right: drop, touchdown, maximum deflection, lift-off, apex. Due to the difference in leg stiffness the biarticular leg is on the ground longer. The biarticular spring deflection is visible in the gap in the biarticular segment at maximum deflection. The biarticular leg adapts its posture, visible in the difference of hip and foot segment angles at maximum deflection.

position of the robot. The biarticular and pantograph leg were dropped from 590 mm hip height. We also measured the input power consumption of the motor driver with a current sensor (ACS713). Data was normalized from hip dropping height to first apex (drop cycle) and averaged over 30 drops and displayed with a 95% confidence interval. Touchdown and liftoff are determined by when the spring-loaded ankle joint starts to deflect. Hip torque is calculated from armature motor current as:

$$x\_{\text{hip}} = i\_{armature} \cdot k\_t \tag{8}$$

where iarmature, is the armature motor current measured on the motor driver and k<sup>t</sup> is the torque constant of the motor. Electrical system input power is calculated as:

$$P\_{el} = U \cdot I \tag{9}$$

where U, is the constant power supply voltage of 24 V, and I is the input current measured from the power supply.

#### 2.3.4. Hopping Experiment

In this section, we provide a realistic locomotion showcase to investigate the behavior of a single hopping leg under a combination of virtual leg axis forces and hip torque. The leg is again constrained to movement in the sagittal plane without trunk rotation by a boom structure.

Both legs use the same gearbox in this experiment to provide enough hip torque for forward locomotion.

A PD position controller calculates the desired current for the low level current controller on the motor driver according to:

$$\begin{split} i\_{\text{motor}\_{\text{desired}}}(\mathbf{t}) &= k \mathbf{p} \cdot (\theta\_{\text{hip}}(\mathbf{t})\_{\text{desired}} - \theta\_{\text{hip}\_{\text{encoder}}}(\mathbf{t})) \\ &+ kd \cdot \frac{d(\theta\_{\text{hip}}(\mathbf{t}) - \theta\_{\text{hip}\_{\text{encoder}}}(\mathbf{t}))}{dt} \end{split} \tag{11}$$

where kp and kd are the proportional and derivative control gains, θhipencoder is the measured hip angle and θhipdesired is the desired hip angle calculated above.

The PD position controller, schematic in **Figure 5**, of the motor driver was tuned to the same gains for both legs. Gait parameters θ0, θ<sup>1</sup> and the gait frequency were hand-tuned for both legs to get stable hopping at 0.99 <sup>m</sup> s with f = 2 Hz for the pantograph leg and 0.95 <sup>m</sup> s with f = 2.2 Hz on the biarticular leg (**Figure 6**). We defined stable hopping, if the robot hopped for trials longer than 2 min, equivalent to ∼= 240 steps. Data was collected 1 min after the robot achieved a stable gait. All data sets were normalized over time from hip apex to apex (step cycle) and averaged over 30 consecutive steps. Average data was displayed with 95% confidence intervals. Touchdown and liftoff were determined by when the springs started to deflect. All future discussions are conducted using the averaged data set to get a representative picture.

FIGURE 5 | Controller diagram for the sine wave position controller used for the forward hopping experiments.

FIGURE 6 | High-speed snapshots of both legs hopping forward for one step cycle from release to hip apex. Pantograph leg top row, biarticular leg bottom row. Depicted are left to right: hip apex, touchdown, maximum deflection, lift-off, second hip apex. The biarticular spring deflection is visible in the gap in the biarticular segment at maximum deflection. Timing differences stem from different duty factor and difference in gait frequency, 2 Hz for the pantograph leg and 2.2 Hz for the biarticular leg. The biarticular leg adapts its posture, visible in the difference of hip and foot segment angles at maximum deflection.

## 3. RESULTS

In this section, we present data and results from the static leg force experiment, the static leg angle torque experiment, the vertical drop test, and the hopping experiment.

## 3.1. Static Virtual Leg Axis Forces

In the pantograph leg knee and ankle angles change equally (**Figure 7**). Because of the parallelogram geometry in the leg's four-bar mechanism, knee and ankle angles are kinematically coupled to be equal. Play in the joints causes the small deviation between the pantograph knee and ankle angle curve. At maximum leg deflection, the pantographs knee and ankle angle deflect by 32◦ . The model prediction fits the data, neglecting the small deviation of ≤ 2 ◦ .

In the biarticular leg, the change in knee and ankle angles are not equal. Because of the biarticular spring, the ankle deflects more than the knee. At maximum deflection the ankle in the biarticular leg deflects by 43◦ , the knee deflects to 17◦ . This first experiment shows, that knee and ankle are not kinematically coupled in the biarticular leg.

## 3.2. Static Leg Angle Torque

Because of the kinetic coupling, both knee and ankle angles in the biarticular leg change under torque (**Figure 8**). The knee on the biarticular leg deflects by 0.85◦ , the ankle deflects by 2.1◦ at 1 Nm. The model shows a reasonable prediction for the angles. The deviation and flat line stem from the hard stop at the knee preventing the knee from over-extension over its resting angle. The hard stop effect is less pronounced in the experimental data, due to material elasticity. It can be seen as a change in slope. The knee model is only valid before hitting the hard stop.

This experiment shows that the biarticular leg can store energy from hip actuation in the biarticular spring. Under the influence of hip torque, the pantograph leg does not change its joint angles.

We argue that the distal elastic element, mimicking the lower leg muscle-tendon structures, acts to the hip actuation like a serial spring. Different from a classical SEA hip actuator, the biarticular spring has components acting in both virtual leg axis and leg angle direction. The ratio of components that act in virtual leg axis or leg angle direction depends on the virtual leg deflection as well as the resting joint angles, and the chosen spring stiffnesses.

These experiments are abstracted from the behavior of a hopping robot. Under only hip torque the data shows that the

FIGURE 7 | Knee and ankle angle changes from resting angles for the static virtual leg axis experiment with both legs. Knee and ankle angles change equally because of the pantograph structure. Knee and ankle change not equally in the biarticular leg because of the additional degree of freedom.

biarticular leg has the ability to store hip actuator energy in its springs. For any given initial posture the leg can adapt its posture and store energy in the springs that can potentially be recuperated.

FIGURE 9 | (A) Mean hip angles for vertical drop tests with 95% confidence levels for 30 drops. Vertical touchdown (left markers) and lift off (right markers) are plotted for pantograph (dotted) and biarticular leg (dashed). The hip angle in the pantograph leg deflects more since the leg only has one DOF. In the biarticular leg, the leg can adapt its internal posture to mitigate the dynamic forces without reflecting them to the femur segment. (B) Knee and ankle angles for vertical drop tests with 95% confidence levels for 30 consecutive steps. Touchdown (left markers) and lift off (right markers) are plotted for pantograph (dotted) and biarticular leg (dashed). In the pantograph leg, the joints deflect symmetrically because of the kinematic coupling. In the biarticular leg, the ankle joint deflects nearly twice as much as the knee joint.

## 3.3. Vertical Drop Test

During stance phase (**Figure 4**), the hip angle in the pantograph leg deflects by 4◦ (**Figure 9A**). Because of the kinematic coupling in the leg, any force reflects into the femur segment and change its angle. Because the biarticular leg has one more DOF, it adepts its posture (**Figure 9B**). By changing the ankle and knee angle the energy is stored in the springs, and the data shows that the hip angle does not change. As the virtual spring induces a torque when the hip angle deflects from its resting angle, the hip torque during stance phase is much higher in the pantograph leg (**Figure 10**). Hip torque is calculated from measured armature current on the motor. The hip torque on the pantograph leg peaks at 2.2 Nm while the biarticular leg peaks at 1.2 Nm. Mean hip torque during stance phase is 0.22 and 0.02 Nm for the pantograph leg and biarticular leg, respectively.

After liftoff, the torque requirement is higher in the biarticular leg. Since the biarticular leg has elastic components in leg angle direction, a force resulting from unloading the joint rapidly reflects into leg angle direction. The virtual leg shoots forward when the two parts of the biarticular spring mount collide due to the hard stop. The collision can also be seen in the provided high-speed video in the **Supplementary Material**. We ignore this reflection effect and do not compensate or utilize it here.

are fixed and the joints do not deflect under hip torque.

adapts its posture forces are reflected less into the femur segment. This results in a lower hip angle change. (B) Mean input power consumption for vertical drop tests with 95% confidence levels. Touchdown (left markers) and lift off (right markers) are plotted for pantograph (dotted) and biarticular leg (dashed). In the biarticular leg, the motor does not need additional power to counteract the force. The biarticular leg requires 81% less motor power to hold the virtual leg vertical.

The duration of stance phase varies between 27% on the pantograph leg and 32% on the biarticular leg. We suspect the difference is due to the higher mass of 29 g as well as the lower global leg stiffness of the biarticular leg.

Because the drop experiment is not a periodic motion, the beginning and end points of the graphs do not match as the leg moves differently for the subsequent lower hops.

As a result of the higher torque requirement, input power shows that the biarticular leg needs less power during stance phase to keep the desired leg posture (**Figure 10B**). After liftoff, the same rise in power that was explained in the hip torque curve is visible. Since oppressing the reflection effect requires high torque at high speed, a drastic rise in power consumption during swing phase is visible. Over the full step cycle, mean power consumption for the pantograph leg is 4.6 and 3.9 W for the biarticular leg. Mean power consumption during stance phase for the pantograph leg is 6.8 W for the pantograph leg and 3.7 W for the biarticular leg. The biarticular leg shows a 46% lower power requirement during stance phase and 15% lower power consumption over the whole drop cycle.

## 3.4. Hopping Experiment

During forward hopping (**Figure 6**) the pantograph and biarticular leg show a similar trend in torque requirements as during the drop experiments before. Hip torque is calculated from measured armature current on the motor. The peak hip

torque is 5.2 Nm for the pantograph leg and 2.4 Nm for the biarticular leg (**Figure 11A**). The mean torque for the pantograph leg is 0.19 and 0.05 Nm for the biarticular leg over the whole step cycle. Mean torque during stance phase is 0.64 and 0.23 Nm for pantograph and biarticular leg respectively. The biarticular leg requires 53% less peak torque and 74% less mean torque.

and mean power is 31% lower than on the pantograph leg.

The previously observed difference in knee and ankle angle between the two legs is also visible during hopping (**Figure 12**). While the pantograph leg deflects both joints the same way due to the four-bar geometry, the ankle deflects more in the biarticular leg. We observe that the knee joints deflect similar in both legs. The biarticular leg's ankle, however, deflects by 45◦ compared to 19◦ on the pantograph leg.

Additionally, the duty factor, the fraction of stance phase over one step cycle period:

$$d\_{duty} = \frac{t\_{Stance}}{T\_{Stepycle}}\tag{12}$$

is much smaller at 31% of step cycle on the pantograph leg than on the biarticular leg where the duty factor is 40%. We assume the duty factor to be higher due to the lower global leg stiffness resulting in an extended stance phase duration as the leg deflects more.

Power requirements during hopping are higher in the pantograph leg than in the biarticular leg (**Figure 11B**). The pantograph leg power peaks at 60 W where the biarticular leg

peaks at 20 W during stance and 30 W during swing because of the reflection effect. Mean input power for the pantograph leg is 14.1 and 9.7 W for the biarticular leg. Mean power requirement on the biarticular leg is 31% lower and peak power requirement is 50% lower. The difference in input power requirement is evident in the cost of transport (COT) (Tucker, 1975),

$$COT = \frac{P\_{in}}{m \cdot \text{g} \cdot \nu} \,, \tag{13}$$

where Pin is electrical input power to the motor driver, m is the robot mass, g is the gravitational acceleration, and v is the forward speed of the robot.

Total COT is calculated using overall input power. The total COT for the pantograph leg is 1.7 compared to the biarticular leg at 1.2. COT when substracting 3 W idle power consumption of the system is 1.3 for the pantograph leg and 0.8 for the biarticular leg. To investigate this further, we calculate the energy stored in the springs and compare the two leg designs.

In the biarticular leg, the overall stored energy is considerably higher than the energy stored in the pantograph leg (**Figure 13A**). The maximum total spring energy in the pantograph leg is 0.45 J vs. 1.56 J for the biarticular leg. Mean spring energy is 0.06 and 0.34 J for the biarticular leg. Total spring energy for the pantograph leg is 82% lower than for the biarticular leg. As we show a higher energy efficiency in the biarticular leg, we conclude that the leg design has a higher recuperation rate. Higher recuperation means the biarticular leg can use the energy stored in its spring more effectively for locomotion.

To get a clearer picture on the biarticular leg, we also plot the individual spring energy contribution to the total energy. The knee spring on the biarticular leg stores roughly the same amount of energy as the single knee spring in the pantograph leg. Peak knee spring energy for the biarticular leg is 0.28 J and

peak biarticular spring energy is 1.3 J. Mean energy stored in the biarticular knee is 0.05 J and mean biarticular spring energy is 0.28 J. The mean energy stored in the biarticular spring is 82% higher.

The maximum power in the springs in the pantograph leg at 12.9 W is 62% lower than in the biarticular leg at 34.4 W (**Figure 13B**). The maximum released power is –11.7 W in the pantograph leg and –28.4 W in the biarticular leg.

The biarticular leg stores more energy in its springs due to the elasticity in both virtual leg axis and leg angle direction. In the vertical drop and the dynamic hopping experiment, the leg recuperates more energy from its springs which reduces the overall required torque and input power.

## 4. DISCUSSION

The placement and functional morphology of elastic elements in legs is an important research question in legged locomotion. In this paper, we show that the biarticular spring, which mimics the elasticity of lower leg muscle tendon structures, has elastic components that can provide series elastic behavior to hip actuation. In a model as well as a static experiment, we show how the biarticular spring enables the leg to deflect its joints at a fixed leg length without changing the hip and foot position. We then show that the additional degree of freedom allows the leg to store energy provided by hip actuation in this elastic element. In a vertical drop test with a virtual spring on the hip, we show that the favorable lower peak torque and power consumption of series elastic behavior do not depend on the motor controller but result from leg mechanics. In the drop experiment, we show that the leg changes its internal posture to adapt to external forces instead of reflecting these forces into the hip actuation. As the hip actuation does not need to compensate for the dynamic loading, no additional torque and power is required, which increases energy efficiency.

Last we show that in a combined load case of torque and virtual leg axis forces, the peak torque and power requirements are lower for the leg with distal series elastic components. By reducing the overall leg stiffness, the leg has a smaller leg length which acts as the lever arm for hip torque to produce a ground reaction force. The higher leg length deflection of 64 mm on the biarticular leg vs. 37 mm on the pantograph leg reduces the mean torque requirement for the leg.

Compared to the vertical drop the biarticular ankle joint during hopping deflects more by 6◦ , even though the hopping height, 490 mm for the pantograph leg and 470 mm for the biarticular leg, is lower than the drop height for the vertical drop. The difference in hip angle stems from the deflection under the additional hip torque to move the leg forward. The higher joint deflection is the result of the combined load case of virtual leg axis forces through dynamic loads and the torque required to vault the leg forward. As expected the biarticular leg stores energy provided by hip actuation in the biarticular spring even under a combined load case of virtual leg axis forces and leg angle torques.

Through the implementation of this elasticity, it is possible to reduce the peak power requirement by 26%, the mean power requirement by 31%, the peak torque requirement by 53% and mean torque by 71% in the hopping experiment.

We show that the biarticular leg with elastic components in leg angle direction possesses the same effects as a series elastic element, namely reduced torque and power requirements. We can, therefore, conclude, that the biarticular leg adds series elastic behavior to the leg. Because the biarticular spring stores 82% more energy we can further conclude that the biarticular spring also reduces the mean power and torque requirements of the leg. The reduced energy requirement shows that in robotic legs leg, compliance in leg angle direction is an equally important design parameter to virtual leg axis compliance.

To put the COT of our design into perspective, **Figure 14** shows the COT values for a selection of robots as well as the regression from Tucker (1975) for animal data over their respective masses. Both the pantograph leg as well as the biarticular leg are below the line for comparable natural runners. We include SPEAR (Liu et al., 2018) as a direct comparison to our monoped hopper. Comparing the COT without base consumption of our biarticular design to SPEAR, the COT of our design is lower at 0.8 than SPEAR at 0.86.

Since power, speed and mass do not scale linearly, as shown by Tucker, we believe that a better comparison than absolute COT numbers, is the comparison to a natural runner of comparable weight, the relative COT. We calculate the relative distance of the biarticular leg's COT to the COT of a model animal of the same weight from the Tucker linear regression.

Comparing the biarticular leg's total COT (including base consumption) to natural runners, the biarticular leg is still below the natural runners line and roughly on the same level as the pantograph leg without base consumption. The relative COT for the biarticular leg is 64% of a natural runner's COT. The relative

as squares, COT values where base consumption (communication, electronics, etc.) is substracted, are shown as diamonds. The COT for the biarticular leg is 64% of the COT of a natural runner with the same weight. All COT values are listed in the Supplementary Material.

COT without base consumption is at 43% of the comparable natural runner's COT. To the best of the authors' knowledge, the biarticular leg shows the lowest achieved relative cost of transport documented for all dynamically hopping and running robots including MIT Cheetah at 68% relative COT.

The only legged robots with a lower relative COT are Cornell Ranger (Bhounsule et al., 2014) and Cargo (Guenther and Iida, 2017). Cornell Rangers COT of 0.19 is 20% of the COT of a comparable natural runner. Cornell Ranger was optimized for COT efficient walking, unlike the here shown dynamic hopping locomotion of the biarticular leg. Cargos COT of 0.1 is 21% of a comparable natural runner. Cargo was designed to run at its natural frequency to increase COT. We exclude Cargo because of its non-practical ground clearance of (Guenther and Iida, 2017, **Figure 12**).

With the results of this paper we create a novel, robotic perspective on the placement and functional morphology of elastic elements in legs. Our research raises the question whether a transfer from the insights from this abstracted model back to biology is possible which has not been shown or discussed in previous research in biology. By showing the same joint deflection behavior under similar load cases, it might be possible, to verify the behavior of the biarticular leg in its natural role models. By finding similar behavior we could then conclude that the anatomy of vertebrate animals is in parts due to the functional morphology shown in this paper. During experimentation, we show a reflection effect that shoots the leg forward at the end of stance phase. While not considering the effect in this study, we will focus our future research on implementing controllers that utilize the effect to further reduce power requirements during the swing phase.

Additionally, we will investigate whether the distal series elastic element increases robustness to perturbations. To follow up the findings in this paper we want to optimize the energy recuperation through an investigation into the effects of posture, segmentation and spring stiffness ratio on the elastic behavior of the leg.

## 5. CONCLUSION

In this paper, we investigate the effects of a distal biarticular elastic element. We show that a bio-inspired distal elastic element has components that deflect in leg angle direction. To characterize the leg we provide a mathematical model, to show the underlying behavior. We then investigate the leg behavior first under virtual leg axis forces. We show that the distal elastic element provides an additional degree of freedom to the leg. In a second step we investigate the leg behavior under only leg angle torque. The second experiment shows that the elastic components in leg angle direction deflect under hip torque and store hip actuator energy. Then we show that the leg can reconfigure its internal posture during a vertical drop experiment. The leg adapts its posture to the loading force, leading to a lower femur deflection. This decreases the power requirement during drop experiments by 46% compared to the leg with only virtual leg axis compliance. The leg angle actuator will therefore require less torque and power to hold the leg during stance. Last we show that the effects investigated in reduced complexity experiments are visible in a realistic monoped hopping experiment with combined leg angle torques and virtual leg axis forces. In the hopping experiment we show

## REFERENCES

that the distal elastic element reduces the power requirements by 31% and the peak torque requirements by 71%. We record a 31% reduced COT of 1.2 for our leg design of 0.9 kg at 1 <sup>m</sup> s . The relative COT of our biarticular leg design is 64% of a comparable natural runner's COT.

## DATA AVAILABILITY

The datasets generated for this study are available on request to the corresponding author.

## AUTHOR CONTRIBUTIONS

FR contributed to the concept, robot design, experimental setup, experimentation, data discussion, and writing. AB-S contributed to the concept, data discussion, and writing.

## ACKNOWLEDGMENTS

We thank the Robotics ZWE under Dr. Fiene as well as the general workshop, the fine-mechanics workshop and the carpentry shop at MPI for providing components for the robots. We thank Felix Widmaier for programming the motor driver interface. We thank the International Max Planck Research School for Intelligent Systems (IMPRS-IS) for supporting the academic development of FR.

## SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnbot. 2019.00064/full#supplementary-material

**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Ruppert and Badri-Spröwitz. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# Cascade Control of Antagonistic VSA—An Engineering Control Approach to a Bioinspired Robot Actuator

Branko Lukic, Kosta Jovanovi ´ c´ \* and Tomislav B. Šekara

School of Electrical Engineering, University of Belgrade, Belgrade, Serbia

A cascade control structure for the simultaneous position and stiffness control of antagonistic tendon-driven variable stiffness actuators (VSAs) implemented in a laboratory setup is presented in the paper. Cascade control has the ability to accelerate, additionally stabilize, and reduce oscillations, which are all extremely important in systems such as a tendon-driven compliant actuators with elastic transmission. Inner-loop controllers are closed in terms of motor positions, and outer-loop controllers in terms of actuator position and estimated stiffness. The dominant dynamics of the system (position and stiffness), composed of the mechanical part and inner loops, are identified by a closed-loop auto-regressive with exogenous input (ARX) model. The outer-loop controllers are tuned on the basis of experimentally identified transfer functions of the system in several nominal operating points for different stiffness values. After the system is identified, a controller bank is generated in which a pair of actuator position and stiffness controllers correspond to a nominal operating point and covers the area surrounding the nominal point for which it is designed. The controllers used are integral-proportional differential (I-PD) and integral-proportional (I-P) controllers, which are a variation of the PID and PI controllers with dislocated proportional and derivative gains from a direct to feedback branch that result to no overshoot for even fast reference changes (i.e., step signal), which is essential for preventing tendon slackening (meeting the pulling constraint). Analytical formulas for controller tuning based on only one parameter, λ, are also presented. Since position and stiffness loops are decoupled, it is possible to change λ for both loops independently and adjust their performance separately according to the needs. Also, the controller structure secures the smooth response without overshooting step reference or step disturbance signal, which make practical implementation possible. After all the controllers were designed, the cascade control structure for simultaneous position and stiffness control was successfully evaluated in a laboratory setup. Thus, the presented control approach is simple to implement, but with a performance that ensures a pulling constraint for tendon-driven actuators as a foundation for bioinspired antagonistic VSAs.

Keywords: antagonistic actuator, tendon-driven actuators, variable stiffness actuators, bioinspired robotics, physical human–robot interaction, position–stiffness control

## Edited by:

Zhao Guo, Wuhan University, China

Reviewed by: Manolo Garabini, University of Pisa, Italy Thomas C. Bulea, National Institutes of Health (NIH), United States

> \*Correspondence: Kosta Jovanovic´<EMAIL_ADDRESS>
Received: 30 April 2019 Accepted: 12 August 2019 Published: 04 September 2019

#### Citation:

Lukic B, Jovanovi ´ c K and Šekara TB ´ (2019) Cascade Control of Antagonistic VSA—An Engineering Control Approach to a Bioinspired Robot Actuator. Front. Neurorobot. 13:69. doi: 10.3389/fnbot.2019.00069

## INTRODUCTION

Robotics has made major strides in recent decades. They began with the deployment of the first industrial robots in a known environment without humans in their immediate proximity. These robots had standard stiff actuators, while elastic deformation in their transmission system was deemed undesirable. The undesirable elasticity could induce oscillations, which engineers attempted to eliminate as early as the mechanical system design stage (Potkonjak, 1988). As industry and medicine developed, and society became more demanding, the need arose for robots and various types of electromechanical devices and drives, whose desired features are highlighted in application areas such as rehabilitation aids, exoskeletons, or service robots. New-generation robots are expected to operate in the immediate vicinity of humans or in collaboration with humans under dynamic conditions where the environment is unknown and changeable. To that end, a new generation of human-like (i.e., bioinspired) actuation is required with the first robotics applications in biologically inspired musculoskeletal humanoids (Diamond et al., 2012; Nakanishi et al., 2013). A number of actuation approaches that resemble properties of muscle system have been developed—tendon-driven and compliant drives that require both side pulling units (antagonistic actuators). For this to be feasible, safe interaction needs to be ensured in some way.

Progressively moving toward the next generation of robotic actuators, the active compliance and stiff actuator shortfalls are overcome with the development of passively compliant actuators. Compliant actuators are often designed as serial elastic actuators (SEAs) with constant actuator stiffness (Pratt and Williamson, 1995; Robinson et al., 1999), or as variable stiffness actuators (VSAs) that, in addition to the position, can control stiffness at the actuator output (Vanderborght et al., 2013; Grioli et al., 2015). Compliant actuators were developed in response to the need to ensure robot compliance and, consequently, safe interaction with the environment and primarily with humans. One of the compliant actuator advantages is that in the event of a collision with the environment, there is no need to wait for the controller to react. Instead, the impact energy is instantaneously absorbed and stored in the actuator's transmission system through deformation of elastic elements. This reduces the effects of interaction forces and ensures safe human–robot interaction. Compliant actuators are more energy efficient because they can utilize the stored energy, which is especially useful in the case of repetitive tasks. When compliant actuators release the stored energy, they exceed stiff actuator performance with regard to peak velocity (Lakatos et al., 2014).

Robots, in general, are designed for specific tasks and specific movements within those tasks. Rigid robots can simultaneously control position and stiffness only with additional feedback loops. The solution for this problem is in the biologically inspired actuator design approach, where actuation is based on the principles of antagonism (antagonistic controlled stiffness), copied from humans and animals and implemented in robots (Migliore et al., 2005; Koganezawa et al., 2006; Jovanovic et al., 2014). Therefore, antagonistic actuators are the bioinspired solution for VSAs, which should bring robot actuation closer to superior human actuation mastered through the evolution. This approach in actuator design improves the quality and variety of robotic movements. Results from Migliore et al. (2005) present loop control of antagonistic VSA, where achieved and commanded position and stiffness have a high level of correlation.

Observing the actuation of living creatures evolved through the centuries, contemporary robot joints are often tendon driven (Mizuuchi et al., 2002; Potkonjak et al., 2011b). The drives are relocated from the joints, and the joints are controlled by tendons wound on reels. The following are some of the implemented tendon-driven robots: the tendon-controlled humanoids Kenshiro, with 64 joint degrees of freedom (DOFs); Kengoro, with 114 joint DOFs without hands (Nakanishi et al., 2012; Asano et al., 2016, 2017); and ECCEROBOT (Wittmeier et al., 2013), which is a fully anthropomorphic, compliantly driven robot. Relocation of the actuators from the arms to the body of the robot reduces the inertia of the arms, which are then lighter and require less energy for control, but elastic elements cause more pronounced oscillations. Some of the challenges are to design an adequate control strategy for this type of actuator, taking into account its non-linearity originating from the condition for the system to have non-linear transmission so that stiffness can be controlled (Van Ham et al., 2009) and to avoid slacking through minimal pre-tensioning of the springs (Potkonjak et al., 2011a). In addition to VSA position and stiffness control in antagonistic actuators such as dynamic feedback control law for input–output decoupling and full state linearization (Palli et al., 2007), or the non-interacting static feedback linearization (Palli et al., 2008), both only validated in simulations, it is possible to apply concepts such as puller– follower (Potkonjak et al., 2011b), where the position and tension forces in the tendons are controlled instead of the VSA position and stiffness in order to preserve tension in tendons. Tendon-driven mechanisms can offer additional control flexibility by exploiting configurations with redundant non-linear elastic tendons, considering conditions under which the joint stiffness is adjustable (Kobayashi et al., 1998).

Newer research is focusing on deriving the control methods for simultaneous position and stiffness control such us feedback linearization for decoupled position and stiffness control with momentum-based collision detection (De Luca et al., 2009), impedance control with static decoupling (Wimbock et al., 2008), or the puller–follower concept where controllability issues are overcome by switching to force–position control (Potkonjak et al., 2011b). However, all abovementioned approaches miss experimental validation, which will make wide use of the presented research. Adaptive neural network control of tendondriven mechanism is experimentally validated (Kobayashi and Ozawa, 2003), but it highlighted stability issues with the presence of the unmodeled dynamics.

This research was motivated by the wish to arrive at a simple, efficient, and robust control system, which can be applied regardless of the type of antagonistic VSA, based on the identification/estimation of the model of the system. The focus of this paper is on the antagonistic tendon-driven type of VSA. The implementation and testing of the proposed cascade control structure with the engineering control approach to an antagonistic actuator is the first step in controlling the broader class of VSAs. The objective is for the control structure to properly control the tendon-driven antagonistic actuator, with no prior knowledge about the model—only some actuator physical parameters that are easy to measure are considered known (i.e., dimensions of motors and actuator pulley radiuses), which will be discussed below. The motivation also traces to earlier activities and papers of the authors in the areas of modeling (Jovanovic et al., 2014) and control (Potkonjak et al., 2011a,b) of antagonistically driven robots. There is some research presenting the successful implemented simulations of position–stiffness control of antagonistic VSA: backstepping control implemented to an antagonistically driven finger with flexible tendons (Chalon and d'Andréa-Novel, 2014), or joint impedance controller with underlying tendon force control that was proposed in Chalon et al. (2011), in both cases, control structure is derived using the analytical model. No widely accepted practical implementation of these methods has been accomplished due to major dependence on the control model, such that the present paper constitutes an upgrade toward the implementation of simultaneous stiffness and position control. Issues in practical implementation for control of antagonistic actuators with the elastic transmission with cascade scheme were pointed out in Lukic et al. (2018). In this paper, the issues are resolved by closed-loop parameter identification and, consequently, the design of robust control scheme with guidelines for parameter tuning. Control structure with a decoupler enables independent control loops for actuator position and stiffness, where controller parameters in each loop are tuned with a variation of only one tuning parameter λ (in general, position and stiffness loop have different λ). Therefore, trade-off between performances and robustness is made by tuning λ. A more detailed explanation of controller tuning and its influence on performances will be presented in the section Antagonistic VSA Cascade Position/Stiffness Control.

In line with all that was previously stated, the presented research introduces a conventional engineering control approach to a bioinspired tendon-driven compliant antagonistic actuator as a concept of new widely accepted safe and efficient robot solution within a human-centered environment. The contribution of this paper is an experimentally validated approach for position–stiffness control of antagonistic VSA. The control is implemented without knowing the exact mathematical model of the actuator (i.e., parameters of non-linear spring characteristics, the transfer function of DC motors, gearbox efficiency coefficient, friction, etc.) but using widely accepted model identification tools for actuator system modeling. Thus, the approach can easily be applied to other types of antagonistic VSAs since the exact VSA parameters are rarely manufactured to fully match its mechanical design due to the complexity of their structure. The paper proposes a control scheme and procedure for controller tuning of antagonistic VSAs that is easy to implement while keeping tendons under tension to prevent slacking. The identified transfer functions present locally linear behavior of the system; thus, controllers are tuned to satisfy stability criteria for linear systems with a certain amount of robustness. The control design is simplified to the selection

In our approach, we presented a cascade control structure for position–stiffness control of antagonistic VSAs, where controller tuning is achieved based on an identified system dynamic. The cascade control structure gives better performance for reference tracking than the classical single-loop control system (Song et al., 2003). Our approach gives a simple procedure for control design through the tuning of free parameter λ. Some papers (Matausek and Sekara, 2011; Sekara et al., 2011; Boskovic et al., 2017) give insight into how controller parameters are tuned as a function of the parameter λ. Our approach requires no prior knowledge of the system model nor higher-order derivatives, and it is easy to implement to a real setup.

The decentralized trajectory tracking (Della Santina et al., 2017; Angelini et al., 2018) is a method where the feedforward component is learned in an iterative procedure to have a good trajectory tracking performance while minimizing the influence of the feedback component. Feedforward control does not affect robot stiffness; thus, natural robot softness is preserved. Shortcomings of this method, compared to ours, are that for every new trajectory, it is necessary to repeat the learning method, which can take a lot of time.

The elastic structure preserving (ESP) and ESP+ control approaches (Keppler et al., 2018) change plant dynamics less than feedback linearization-based control and aim to minimize the dynamic shaping and preserve the elastic structure of the system. The limitation of this approach is that the model-based control law requires higher-order derivatives of link positions, while in our case, a control law is model free and exploits only information of actuators and link positions.

Section Antagonistic VSA Drive: Prototype and Modeling of the paper describes the general structure, mathematical model, and physical implementation of the laboratory setup of the bioinspired antagonistic VSA, as well as the way in which the system was identified. The cascade control structure and controller tuning are discussed in the section Antagonistic VSA Cascade Position/Stiffness Control. The section Experiment Results contains the experimental results, and the section Conclusions summarizes the conclusions and directions of future research.

## ANTAGONISTIC VSA DRIVE: PROTOTYPE AND MODELING

**Figure 1A** shows the mechanical configuration of an antagonistic VSA developed in the ETF Robotics Laboratory of the University of Belgrade. The laboratory setup comprises (1) a link; (2) a nonlinear compression springs with a quadratic characteristic; (3) two Dunker GR42X25 DC motors with Dunker SG45 gearbox, ratio 15:1; (4) a sliding mechanism for pre-tensioning; (5) two TAL501L force sensors made by HT Sensor Technology, which measure tendon tension; (6) an ISC3806-003E-1000BZ3- 5-24F incremental rotary encoder for actuator position tracking; and (7) two DHC40M6-2000 incremental rotary encoders for

position tracking at the gearbox outlet; (8) an end-position magnetic sensor; (9) magnets; (10) a safety circuit; (11) a power sources; and (12) a pedestal.

The structure of the actuator is attached to a plate acting as a pedestal of the apparatus. The antagonistic drive itself is positioned on sliders used to adjust spring pretension. This also facilitates spring replacement and pre-tensioning, if springs of different lengths and characteristics are to be used. The apparatus was sized such that the link radius at the gearbox outlet is r<sup>m</sup> = 7.5 mm and the actuator radius r<sup>j</sup> = 75 mm. Signals are acquired by a National Instruments NI-PCIe6323 acquisition card with analog outputs for motor control.

**Figure 1B** is a block structure of the antagonistic VSA. Force (dashed orange lines) and actuator and motor position (dashed blue lines) measurements are interfaced with the computer via the NI-PCIe6323 acquisition card. Computed control signals are forwarded via analog outputs of the acquisition card to the safety module. The safety module operates independently and acquires force sensor data and end-position magnetic sensor data (dashed red line). If all signals are within prescribed boundaries, the control signals are forwarded to the motors, and if the actuator position or tension exceeds the permissible level, the motors immediately shut down to prevent damage to the apparatus. The control structure was implemented in a Matlab/Simulink Real-Time environment.

## MATHEMATICAL MODEL OF ANTAGONISTIC VSA

The tendon-driven antagonistic VSA operates in such a way that motor rotation around outlet shaft groves causes the tendons to wind/unwind and, thus, compress/decompress the springs. Simultaneous compression increases actuator stiffness, while winding of one tendon and unwinding of another changes the position. In order to control antagonistic actuator stiffness, the stiffness in the transmission mechanism (in this case spring) between the motor and the link needs to be non-linear, with a monotonously incremental force relative to extension (Van Ham et al., 2009). The pair of springs used in the apparatus were custom-made. The requirements were that they be identical and have a monotonous growing force/elongation characteristic.

Spring elongations ∆l<sup>A</sup> and ∆l<sup>B</sup> of the antagonistic actuator are defined by:

$$
\Delta l\_A = r\_m \theta\_A - r\_j q \tag{1}
$$

$$
\Delta l\_B = r\_m \theta\_B + r\_j q \tag{2}
$$

where the positions of the actuator, motor A and motor B are q; θA, and θB, respectively.

Force/elongation characteristic can be represented in a polynomial form as:

$$F\_A = \sum\_{i=0}^{n} k\_i \Delta l\_A^i \tag{3}$$

$$F\_B = \sum\_{i=0}^{n} k\_i \Delta l\_B^i \tag{4}$$

where F<sup>A</sup> and F<sup>B</sup> denote forces that compress the springs, where ki for i = 1 ... n, polynomial coefficients, and n is the order of the polynomial.

The schematics of the variable stiffness antagonistic actuator is depicted in **Figure 2**, while a model that describes actuator dynamics and the dynamics of motors A and B is given in Equations (5) to (7):

$$J\_{\dot{\jmath}}\ddot{q} + B\_{\dot{\jmath}}\dot{q} = \varphi\left(\theta\_A, \theta\_B, q\right) \tag{5}$$

$$J\_m \ddot{\theta}\_A + B\_m \dot{\theta}\_A + \phi \left(\theta\_A, q\right) = \ \mathfrak{r}\_{mA} \tag{6}$$

$$J\_m \ddot{\theta}\_B + B\_m \dot{\theta}\_B + \psi \left(\theta\_B, q\right) = \text{ } \mathfrak{r}\_{mB} \tag{7}$$

where J<sup>j</sup> and J<sup>m</sup> are actuator and motor inertia, respectively. B<sup>j</sup> and B<sup>m</sup> symbolize the equivalent viscous friction in the actuator and motors, respectively. The torques of motors A and B are denoted by τmA and τmB, respectively, and can be represented as:

$$
\pi\_{mA} = \mu N \frac{K\_{mu}}{Ls + R} U\_A \tag{8}
$$

$$
\pi\_{mB} = \mu N \frac{K\_{mu}}{Ls + R} \, U\_B \tag{9}
$$

where L and R are induction and resistance of a motor, respectively. Kmu; N, and µ symbolizes the motor torque constant, gearbox reduction ration, and gearbox coefficient of efficiency, respectively. U<sup>A</sup> and U<sup>B</sup> are voltage inputs on motor A and motor B, respectively. The s is Laplace transformation operator.

The equivalent actuator torque and parts of the actuator torques originating from the elastic springs connected with

motor A and motor B are ϕ; φ, and ψ, respectively (shown in Equations 10–13):

$$
\phi \left( \theta\_A, q \right) = r\_m F\_A \tag{10}
$$

$$
\psi\left(\theta\_B, q\right) = r\_m F\_B \tag{11}
$$

$$\varphi\left(\theta\_A, \theta\_B, q\right) = \frac{r\_j}{r\_m} \left(\phi\left(\theta\_A, q\right) - \left|\psi\left(\theta\_B, q\right)\right|\right) \tag{12}$$

by combining Equations (10) to (12), Equation (13) for actuator torque follows:

$$
\varphi\left(\theta\_A, \theta\_B, q\right) = r\_j \left(F\_A - F\_B\right) \tag{13}
$$

Actuator's stiffness S is defined by Equation (14).

$$\mathcal{S} = -\frac{\partial \varphi \left(\theta\_A, \theta\_B, q\right)}{\partial q} \tag{14}$$

The equilibrium position of the actuator q<sup>e</sup> for the symmetrical system shown in **Figure 1**, based on known geometric relations and symmetry, is:

$$q\_e = \frac{r\_m \left(\theta\_A - \theta\_B\right)}{2r\_j} \tag{15}$$

The expression for actuator stiffness is derived by combining Equations (3) and (4) with Equations (13) to (15):

$$S = 2r\_j^2 \sum\_{i=1}^n \frac{i}{2^{i-1}} k\_i r\_m^{i-1} \left(\theta\_A + \theta\_B\right)^{i-1} \tag{16}$$

## PARAMETER ESTIMATION

In order to make the approach versatile and applicable to a different kind of antagonistic VSAs, several main actuator characteristics have to be estimated. Initially, each individual drives and gearing have to be identified as a core of the internal loop. Elastic transmission elements determine the actuator dynamics, and consequently, they have to be well-estimated for the control purposes as well as for the evaluation of the actuator performances. Finally, for the sake of reliable and robust position and stiffness control, outer loops that comprise inner loops, elastic transmission, and actuator mechanical design have to be identified.

Fast system dynamics is determined by the electric drive and its gearbox. The goal of the control approach is to accurately and promptly control antagonistic drive positions so the outer position and control loops are preserved. In order to tune the inner loop, the parameters of the assumed geared motor transfer function G<sup>m</sup> (s) were identified. The transfer function was assumed to be a second-order transfer function without time delay, where voltage is the input and motor position the output. The parameters were estimated by first estimating those of the motor speed transfer function, without payload, described as a first-order transfer function without time delay, which is a common approximation in engineering. In general, velocity/voltage transfer function of the motor gearbox set is a second-order transfer function with two real poles on the left side of the complex plane, where one pole is determined by electric characteristics and the second pole is determined by mechanical characteristics. The mechanical component of the system is significantly slower than electrical; thus, its dynamic behavior is dominant and transfer function can be estimated as a first-order transfer function (Leonhard, 2001; Ogata, 2009). Gain K<sup>m</sup> and time constant T<sup>m</sup> were estimated from the input specified as a step signal and from the motors' velocity step response. Then, an integrator was added to the resulting transfer function because the position is the integral of speed. In accordance with the adopted first-order transfer function, gain K<sup>m</sup> is estimated as the ratio between the input voltage and output velocity, while time constant T<sup>m</sup> presents the time required to motor reach 63.21% of the steady-state value. Identification experiments were executed for several input amplitudes. Parameters K<sup>m</sup> and T<sup>m</sup> were estimated as mean values of all experimental results, while there were no significant deviations from the mean values. Hence, the motor transfer function G<sup>m</sup> (s) with estimated parameters is:

$$G\_m \left( s \right) = \frac{K\_m}{s \left( T\_m s + 1 \right)} \tag{17}$$

where K<sup>m</sup> = 2.86 is the estimated gain factor and T<sup>m</sup> = 0.22 is the estimated time constant.

Actuator dynamics and its stiffness are determined by elastic elements used in the transmission. In order to estimate actuator stiffness according to (16), spring parameters need to be estimated as well. If actuator elastic elements/springs are changed, spring parameters must be estimated again. To accurately determine the spring coefficients, the spring characteristic was captured by measuring forces and spring elongations at different loads. The fits of the spring function for 2nd-, 3rd-, and 5th-order polynomial satisfying the least square error are shown on the top plot, while estimation errors are shown on the bottom plot in **Figure 3**. As expected, with polynomial order increase, estimation error decreases. From the observation of the measurement and polynomial approximations, the 2nd-order produces error <0.85 N and the higher polynomial order do not decrease error significantly. For controlling the tendon-driven actuators, there is always some minimal tension required in tendons (non-linear springs); thus, the amplitude of errors will be relatively small in comparison with the amplitudes of working forces (Potkonjak et al., 2011b). The 2nd-order polynomial model presents the compromise between the accuracy on the one side and its easy implementation on the other side. In cases when this error amplitude is not negligible, the higher-order model must be used. The resulting coefficients are k<sup>0</sup> = −2.498 N, k<sup>1</sup> = 331.49N/m, and k<sup>2</sup> = 8543.8N/m<sup>2</sup> .

Finally, actuator performances depend on its electromechanical design as a whole. **Figure 4** shows the structure of the system whose dominant dynamics is identified. The system is composed of the mechanical part of the antagonistic VSA and motor position inner loops. The

system inputs are reference motor positions, and the outputs are actuator's stiffness and position. The transfer functions of real systems, which are continuous in nature, based on periodic discrete measured data (from computer or microcontroller), can readily be represented by a discrete equivalent such as the ARX model (autoregressive model with exogenous inputs) (Jansson, 2003). Identification was undertaken in a closed loop as described in Mataušek and Šekara (2014) and Hjalmarsson (2005), and it is shown in **Figure 5**. Close loop identification is required to reduce the effect of sensor noise and avoid output drift (Skogestad and Postlethwaite, 2001), which cannot be achieved through the open-loop identification.

force/elongation characteristic for 2nd, 3rd, and 5th order polynomial.

**Figure 5** depicts the force control loop used only for the identification of outer loop transfer functions—actuator position [G<sup>q</sup> (s)] and stiffness [G<sup>S</sup> (s)]. The control loop for identification consists of controller Cp(s) and transfer function G<sup>p</sup> (s) that include both actuator position and stiffness transfer function: G<sup>q</sup> (s) and G<sup>S</sup> (s). Transfer function Gp(s) includes the mechanical design of the antagonistic actuator with elastic transmission elements and inner loops as shown in **Figure 4**. Controller Cp(s) controls the tendon tension force in the outer loop around a nominal point. For identification purposes, controller Cp(s) in the outer loop does not need to be optimally tuned but stable. In general, the feedback need not to be same as the output variable, which is being identified, it only needs to be stable in order to avoid output drift and keep the system around a nominal setpoint. Also, closed-loop identification is carried out to reduce the influence of sensor noise and therefore improve identification. Based on the inputs and the outputs of the system Gp(s), dominant dynamics of the actuator position and stiffness is identified.

The system shown in **Figure 4** has two inputs: motor positions θ<sup>A</sup> and θB, and two outputs: actuator position and stiffness q

motor position loops.

and S. Given that the system is symmetrical and that both inputs contribute equally to the output, identification is undertaken with respect to only one motor, while the other is kept still. Without loss of generality, it is deemed that θ<sup>B</sup> = 0.

Since the system is non-linear, it was identified in the vicinity of several operating points, for tension forces of 10, 20, 30, and 40 N, which correspond to the estimated stiffnesses of 8.24, 10.55, 12.43, and 14.06 Nm/rad, respectively. The stiffness range is limited by the properties of the actuator design and compression springs, which are fully compressed at 50 N. However, the proposed control approach is invariant to the stiffness range.

While the outer loop keeps the system at a nominal tension force, a low-amplitude PRBS (pseudorandom binary sequence) signal is added to the input-desired motor position, just enough for its effect to be visible at the output and distinguishable from sensor noise (Polóni et al., 2008). The PRBS with a sampling period of 1 ms was applied. The one sequence of the PRBS signal consists of binary signals generated under defined rules (rules for generating the PRBS signal refer to the shift registry and polynomial rules based on the signal length–number of samples within one period) (MacWilliams and Sloane, 1976), with the property of constant amplitude spectrum that excites the system on all frequency equally.

Based on the measured input of the system Gp(s) (motor A reference position of θAref) and output (actuator position, q; stiffness, S; and tendon tension force, F), the ARX model was used to identify the dominant dynamics for actuator position and stiffness at each of the nominal values. The dominant dynamics is the lowest-order polynomial transfer function whose response adequately mimics the real response of the system. **Table 1** shows the estimated transfer functions of the system for position G<sup>q</sup> (s) and stiffness G<sup>S</sup> (s) of the antagonistic actuator at various nominal points obtained through identification, where G<sup>q</sup> (s) and G<sup>S</sup> (s) are grouped in G<sup>p</sup> (s) from **Figure 5**.

Results are divided into two groups: first- and third-order estimations of the transfer function. All estimated function of the third order has one real pole and one pair of conjugated complex poles. A pair of conjugated complex poles produces some oscillatory dynamics, but the real pole is at much lower frequencies; thus, this dynamics is dominant and system transfer function can be approximated as the first-order transfer function.

The predominance over the oscillatory dynamics could be explained by the existence of the inner loop controller, which shapes on overall input/output dynamics.

ANTAGONISTIC VSA CASCADE POSITION/STIFFNESS CONTROL

The control structure is realized as a cascade control structure for simultaneous position and stiffness control, which is shown in **Figure 6**. Compared to a single feedback loop and having in mind that the inner and outer loops rely on the same control elements (DC motors), the cascade structure contributes to better performance of the system because the inner loop is designed to act locally and to always be faster than the outer loop (Skogestad and Postlethwaite, 2001). Furthermore, due to its faster dynamics, the inner loop adjusts DC motor behavior to tackle non-linearities (including friction) ahead of the outer loop, thereby minimizing negative effects of the tendon slackening, which is a crucial control issue in tendon-driven actuators (Lukic et al., 2018). The cascade structure removes a disturbance more effectively, thus achieving a faster response of the entire system and reducing oscillations, all of which are of major importance for tendon-driven robots to enable practical implementation. Thus, since the inner loop makes more rapid adjustments, the outer loop can be tuned more conservatively (Song et al., 2003). This is of particular importance since theoretical control approaches require smooth position and stiffness generation to the high-order derivatives.

The cascade control structure shown in **Figure 6** comprises (1) a mechanical system with the antagonistic drive; (2) two motor position inner loops; (3) actuator position outer loop; (4) actuator stiffness outer loop; and (5) a static decoupler that computes reference motor positions for the desired actuator position and stiffness.

All controllers used in the inner and outer loops are integralproportional differential (I-PD) or integral-proportional (I-P) controllers, with a filter (which filters out the sensor noise and ensures causality of the controller transfer function) (Åström and Hägglund, 2006; Shamsuzzoha and Lee, 2007). Introduced I-PD/I-P controllers facilitate appropriate parameter tuning, such that the performance of the closed-loop systems is satisfactory in terms of bandwidth, trajectory tracking, robustness, and disturbance rejection. Note that the bandwidth of the system is limited, and system response cannot be accelerated more than the mechanical properties allow.

Static decoupler D (**Figure 6**) reproduces the actuator position and stiffness at the output of the outer loop controllers qcont and Scont in reference motor positions θAref and θBref . It is based on the estimated relations for the position (15) and secondorder polynomial for stiffness (16) of the actuator in the state of equilibrium.

$$
\theta\_{A\_{ref}} = \frac{r\_{\circ}}{r\_m} q\_{cont} + \frac{S\_{cont} - 2k\_1 r\_{\circ}^2}{4k\_2 r\_{\circ}^2 r\_m} \tag{18}
$$

$$\theta\_{B\_{ref}} = \frac{-r\_j}{r\_m} q\_{cont} + \frac{S\_{cont} - 2k\_1 r\_j^2}{4k\_2 r\_j^2 r\_m} \tag{19}$$

Equations (17) and (18) can be written in matrix form as:

$$
\begin{bmatrix} \theta\_{A\_{ref}} \\ \theta\_{B\_{ref}} \end{bmatrix} = D \begin{bmatrix} q\_{cont} \\ \mathcal{S}\_{cont} \\ 1 \end{bmatrix} \tag{20}
$$

$$D = \begin{bmatrix} \frac{r\_j}{r\_m} & \frac{1}{4r\_j^2 r\_m k\_2} & \frac{-k\_1}{2k\_2 r\_m} \\ \frac{-r\_j}{r\_m} & \frac{1}{4r\_j^2 r\_m k\_2} & \frac{-k\_1}{2k\_2 r\_m} \end{bmatrix} \tag{21}$$

where matrix D is the system decoupler and 2k1r 2 j is the minimum actuator stiffness derived from the estimated spring model under the pulling constraint.

In cases when stiffness is presented as higher-order polynomic or a complex non-linear equation (Catalano et al., 2011), it is not possible to present an analytical solution for decoupler function. Thus, the mapping between motor positions and actuator stiffness can be achieved with a lookup table or using some computing tool such as a neural network (Lukic et al., 2016).

The following part of this section introduces guidelines for inner and outer loop control design using complementary sensitivity function shaping. Following the control approach presented in Matausek and Sekara (2011), Sekara et al. (2011) and Boskovic et al. (2017), dynamic behaviors of the closedloop systems are directly affected by the parameter λ as an only tuning parameter. Therefore, complementary sensitivity function shaping enables easy control design of the controller, its easy adaptation to the changes in the system, and its desired behavior by changing parameter λ. Lower values of λ mean faster closedloop dynamics, but higher gains and, therefore, higher amplitude

peaks in the control signal, and lower λ will provide slower system dynamics with longer but lower amplitude control signals. Some papers (Matausek and Sekara, 2011; Sekara et al., 2011; Boskovic et al., 2017) give insight into how controller parameters are tuned as a function of the parameter λ and its influence on the closed-loop dynamics. From these papers, rules for controller parameter tuning and closed-loop dynamics behavior shaping even for higher-order systems with parameter λ can be extracted. However, for the purpose of this paper, the only dynamical behavior of the first- and the second-order systems is required. The procedure for the derivation of controller parameters is given in **Appendix A**.

## TUNING OF INNER LOOP CONTROLLERS

This section introduces guidelines for tuning of the inner control loop–motor position control. Taking the parameters of the transfer function from Wittmeier et al. (2013), a robust I-PD controller was designed to reject disturbances and minimize the integral of the absolute error according to existing analytical formulas for several typical second-order systems, introduced and mastered by the third author (Matausek and Sekara, 2011; Sekara et al., 2011; Boskovic et al., 2017). **Table 2** shows the controller parameters as functions of system (motor) parameters and tuning parameter λ, whose tuning affects system performance and robustness.

The controller was designed to satisfy the desired sensitivity function msm, defined as ms<sup>m</sup> = max ω 1 1+Gm(jω)Cm(jω) , where C<sup>m</sup> jω is the controller that regulates the operation of the motor. The actuator range used in the experiments is 180 deg, which corresponds to 10 times wider motor range according to joint/motor pulley radius [see Equation (15)]. Another constraint in the experiments is the stiffness range (8–15 Nm/rad). The constraints are caused by the mechanical design of the actuator. Maximal disturbance introduced as an additional spring tension in robustness testing is 50 N, which corresponds to 0.375 Nm torque disturbance at the gearbox shaft. The value ms<sup>m</sup> = 1.4 was

TABLE 2 | I-PD and I-P controller analytical formulas for identified motor transfer function and first-order transfer function without time delay (GS(s) and Gq(s)).

determined experimentally; in such a way that it is a good tradeoff between trajectory tracking performance and robustness. The desired value of ms<sup>m</sup> for motor transfer function identification is achieved at λ = 0.0854. The controller parameter k<sup>r</sup> can be tuned in the (0 − kp) range, and the value of parameter b is in the (0– 1) range. If b is higher, when the reference changes, the system will respond more quickly but the overshoot will also be larger. Without loss of performance in terms of disturbance rejection and robustness, b = 0 and, hence, k<sup>r</sup> = 0 is selected. Rejection of the overshoot is an essential requirement in the inner loop, which should prevent oscillation in compliant tendon-driven actuation. Since the system is symmetrical, both motors have the same controllers.

## TUNING OF OUTER LOOP CONTROLLERS

This section introduces guidelines for tuning of the outer control loops–actuator position and stiffness control. For transfer functions of the first and third orders for position and stiffness given in **Table 1**, in each case, the dominant behavior is welldefined with its first-order transfer function; thus, the dominant dynamics can be represented as a first-order transfer function without time delay, with static gain K and time constant T.

Given that the transfer functions were identified as first-order transfer functions without time delay, the outer loop controllers were tuned like I-P controllers according to Boskovic et al. (2017). The I-PD/I-P controllers are variations of the PID/PI controllers, where the proportional and derivative gains from the direct branch are dislocated to the feedback branch, and it will reflect to no overshoot even for fast reference changes (i.e., step signal); thus, it will keep tendons under tension and prevent the slacking. Here, λ is again a tuning parameter that can be used to tune the response rate. The advantage of these controllers is that by tuning just one parameter, it is possible to make a balance between performance and robustness. The λ is a monotonic parameter, which means that a lower value always means a lower closed-loop system time constant and, therefore, faster response, but less robustness. For higher values of λ, results are opposite. Another benefit of centralized controller tuning is the avoidance of actuator saturation, which could deteriorate pulling constraint and performances of a tendondriven system, especially if it comprises elasticity. To that end, a decrease in λ leads toward actuator limits and uses of full actuator range. Online changing of λ allows the implementation of gain scheduling depending on desired performances and response rate for different operating points. If the identification process reveals a more complex dominant dynamic, with multiple poles, the analytical formulas for the controller parameters can be derived as described in Matausek and Sekara (2011). **Table 2** shows the analytical formulas for I-P controller tuning. To achieve a satisfactory response rate without overshoot, λ = 0.2 and b = 0 were adopted for all position and stiffness controllers and present initial tuning.

Given that the decoupler introduces a gain between the outer loop controller output and the reference positions of the motor, in order to retain the performance of the designed controllers, gains kp; k<sup>i</sup> , and k<sup>r</sup> need to be divided by the gains introduced by the decoupler. According to the decoupler values from (20), actuator position gains should be divided by <sup>r</sup><sup>j</sup> rm and the actuator stiffness controller gains by <sup>1</sup> 4r 2 j rmk2 .

In the case of non-linear systems, due to different dynamics in different operating points, controllers can be designed for a linearized model, for each of the operating points of a set, thus forming a controller bank or set. The set of controllers can be used to control the system in such a way that each of the controllers would be tasked with a part of the operating range, for example, controller 1 would cover stiffness from S<sup>1</sup> to S2, controller 2 from S<sup>2</sup> to S3, controller 3 from S<sup>3</sup> to S4, and so on. In this research, bumpless switching between controllers is used (the integral component takes over the control value while switching between two different controllers within the controller bank). Another application is to use a certain type of gain scheduling, where controller gains are computed as a linear combination of the two nearest controllers, depending on the nominal for which they were designed and on the desired value.

It is of most importance to ensure a pulling constraint and avoid tendon slacking. Therefore, outer loops could be easily prioritized to ensure pulling constraint by making position control slower than stiffness control, which guarantees no slacking and, thus, allows validation on the laboratory setup. This can be achieved by adjusting the parameter λ separately for each of the outer loops, since the response time is proportional to the parameter λ for each loop individually (Sekara et al., 2011). Thus, this parameter dictates the response dynamics by trading off between the robustness and performance.

## EXPERIMENT RESULTS

Performances of controllers are verified in two experiments. The first experiment is position and stiffness tracking while their references are changing separately. In this scenario, actuator stiffness is estimated based on (13). In the second experiment, actuator stiffness/compliance control is validated by tendon force measurements, and therefore, stiffness is estimated directly from actuator torque and position measurement (12).

**Figure 7** presents the results of the first experiment. **Figures 7A,B** show actuator position and stiffness tracking, where the controller was designed for a nominal operating point of S = 10.55 Nm/rad. First, the position reference was specified as a step signal sequence in the range from –π/3 to π/3, while stiffness remained at 10 Nm/rad, and then, the position was held constant while stiffness was varied as a pulse signal in the range from 8 to 14 Nm/rad. **Figure 7C** depicts the control signals of the motors. As expected in the case of actuator position tracking, one motor winds the tendon, while the other unwinds it, as evidenced by the control signals that have the same amplitude with different signs. When stiffness changes, both motors have identical control signal amplitudes because they simultaneously compress and decompress the springs, such that the results are as anticipated. **Figures 7D–F** show the results when the actuator position and stiffness references were changed consecutively, without time delay, so that both variables were stationary before the reference changes. The results depict position and stiffness tracking, as well as system decoupling, all together with the control effort.

In the second experiment, actuator stiffness tracking is evaluated by measuring tendon forces and actuator position. To test the stiffness, the reference actuator position was set at 0 and the reference stiffness at 10 Nm/rad. In this case, only the stiffness controller is used in the outer loop to achieve compliant behavior. The actuator position control is in the open loop, where regulation relies on the geometrical relation (14) implemented through the static decoupler D. Therefore, outer position loop is not aware of the externally applied torque. Trajectory tracking performance will depend on the accuracy of the motor and actuator radii as geometric actuator parameters, which are easy to measure. The torque that acts on the link in the state of equilibrium is equal to 0, such that disturbance Fext, which is a manually entered force that acts tangentially on the actuator link, was measured as the difference between the forces that act on the actuator, measured in the stationary state:

$$F\_{\text{ext}} = F\_A - F\_B \tag{22}$$

Instead of a derivative (12), can be represented by means of a linear approximation, as the quotient of the torque change and angle change:

$$\mathcal{S} = -\frac{\Delta\Phi}{\Delta q} \tag{23}$$

Note that externally applied torque is related to externally applied force by a scale factor r<sup>j</sup> , so actuator stiffness will be estimated from the applied force, which could be directly measured by sensors. Since the nominal angle is 0, and the torque acting on the actuator is r<sup>j</sup> (F<sup>A</sup> − F<sup>B</sup> ),

$$S = -\frac{r\_j F\_{ext}}{q\_m} \tag{24}$$

holds for (21), where q<sup>m</sup> is the instantly measured actuator position (position displacement due to external force Fext). Therefore, the externally applied force could be estimated based on measured actuator position and commanded stiffness as follows:

$$F\_{expected} = -\frac{Sq\_m}{r\_j} \tag{25}$$

Given that the actuator radius r<sup>j</sup> is constant and that the stiffness is held constant, the correlation between the force and the position change is also constant. **Figure 8A** shows measured actuator positions q. The external force Fext, which is the difference between measured forces F<sup>A</sup> and FB, and the expected force Fexpected based on (23) for the measured position and commanded stiffness are depicted on **Figure 8B**. It is apparent that the shape of all the graphics is the same and that there is a high level of correlation, but that there are minor deviations between measured and expected forces, as shown in **Figure 8C**. The amplitude of the differences is relatively small. The differences occur because (22) and (23) are only linear approximations and the stiffness expression is derived according to (13), which is based on spring parameter approximation. The computed root mean square error (RMSE) for the signal from **Figure 8C** is 2.239 N. This may seem like a relatively high value, but considering that used force sensors have a time delay (10–15 ms), and expected force is computed based on encoder measurements that do not have a time delay, these two signals are not aligned and therefore measured errors and obtained RMSE are presented larger than they really are. Similarly, the two springs are not exactly identical and that introduces additional errors into the modeled system. This experiment demonstrates the practical evaluation of the antagonist VSA stiffness tracking.

## CONCLUSIONS

Looking toward future robot actuators that comprise intrinsic compliance for safety and energy efficiency, tendon-driven actuation for reducing robot inertia, and antagonistic configuration with non-linear transmission for enabling

bidirectional moves and variable actuator stiffness/compliance, this paper elaborates design and control of tendon-driven compliant antagonistic VSA that can be applied on all antagonistic VSAs. Since the antagonistic design of VSA, mastered by nature through evolution, presents a foundation for moving of humans and mammals in general, this is undoubtedly a design of future robot actuators, which aims to work in the human-centered environment. This research introduces a conventional engineering control approach based on cascading structure to a bioinspired compliant antagonistic actuator.

A cascade structure for simultaneous position and stiffness control of an antagonistic VSA was presented and experimentally tested in the paper. A cascade structure contributed to the stability of the system because a proper inner-loop controller design reduced oscillations at the output from the system, which can occur especially in mechanical systems that feature elastic transmission. Robust controllers were designed for motor positions on the basis of experimental identification of the gearbox motor transfer function. In addition, locally linearized actuator position and stiffness transfer functions were identified with motor position inner-closed loops, such that the controller dynamic was included in the identification. The linear system control theory for controller tuning was applied. A static decoupler was designed based on an estimated spring model and a mechanical design of the actuator.

The advantage of this approach involving system identification is that it requires no previous knowledge about the model of the system and no information about system parameters, such as motor and actuator friction and inertia, gearbox efficiency, force sensor mass, etc. Another advantage is that it is not necessary to measure or control motor torques. Control is achieved via voltage, and the connection between voltage and torque is included in the dynamic that is being identified. Only a part of the system model is required, related to actuator stiffness and position (13) and (14), as well as (15), whose relations were determined experimentally. Consequently, a similar control system design principle can be applied to different types of antagonistic VSAs.

The structure of the I-PD/I-P controllers that were used is selected to satisfy pulling constraint of tendon-driven actuators. The controllers were designed with the desired robustness, such that apart from the dynamics disregarded in the identification process due to their minor effect, and in the case of non-modeled dynamics occurring during continuous operation of the system due to a change in any of the system parameters, the controllers will maintain stability and good trajectory tracking performance. The main feature of the applied controllers is the easy tuning of the system performances as a trade-off between robustness and speed/performance, by changing only one parameter λ. Therefore, by changing λ online, the great flexibility for the system adaptation to different scenarios and change of the robustness/performances for different operating points are feasibly and easy. The robustness is of great importance in tendon-driven compliant systems so the pulling constraint is successfully tackled.

In order to control antagonistic VSAs more appropriately, especially when a broader stiffness range is available, the controller bank is used. Each controller in the bank is tasked with covering a span around the nominal point for which it was designed, while bumpless switching between controllers is exploited.

Performances of controllers are verified in two experiments: simultaneous position and stiffness tracking while their references are changing separately while actuator stiffness is estimated from the model, and actuator stiffness/compliance control validated by stiffness calculated from force measurements directly.

Extending this control structure to the other types of VSAs requires modification of the current approach. In general, a VSA is a two-input–two-output system. If the actuator stiffness and position are not coupled, the actuator can be observed as two single-input–single-output systems that can be adjusted independently. In a more complex case, if the actuator stiffness and position are coupled, a modification of the proposed approach is needed. The first difference to the presented approach is the identification procedure since input/output dynamic behavior is not symmetric, as it is the case of the antagonistic structure. The same cascade control structure can be kept, but the identification must be conducted from both input channels. To enable the generalization approach presented in this paper to the wider class of the VSAs, it is necessary to overcome the problem of the asymmetric input–output transfer function of VSAs. This can be achieved by introducing the dynamic decoupler that will make outer loop control decoupled from the system. Dynamic decoupler is computed based on locally identified transfer functions (Nordfeldt and Hägglund, 2006), for both input–output channels. For this case, outer loop controllers must be tuned according to the dynamic decoupler values. Now, a new problem arises, dynamic decoupler is locally valid because it is computed according to the locally identified transfer functions. The goal of this paper was to find a simple and yet efficient method for controlling antagonistic actuator as bioinspired VSA; thus, the generalization to the wider class of VSA control will

## REFERENCES

Åström, K. J. T., and Hägglund (2006). "Chapter 6: PID design," in Advanced PID control, ISA-The Instrumentation, Systems, and Automation Society, 2006 (Durham, NC), 158–225.

be done in future work, with a focus on how to project robust controllers with a bank of dynamic decouplers.

Further research will examine additional system identification methods, which could lead to more flexible actuator identification procedures and therefore facilitate the broader use of the presented approach to a wider class of VSAs. Also, the influence of different non-linear transmission elements will be evaluated. To increase robustness of the actuator, different approaches to disturbance rejection would be considered (Guo et al., 2018; Yuan et al., 2019). Moreover, the upgrade of the presented results will take into account the inherent asymmetry of the system because the two motors and two springs are not exactly identical. In that case, the elastic elements of arbitrary characteristics can be modeled as higher-order polynomials, where it is not possible to analytically implement a decoupler. Instead, a non-conventional mapping approach would be needed, such as that based on neural networks and initially validated by the authors in Lukic et al. (2018).

## DATA AVAILABILITY

The datasets generated for this study are available on request to the corresponding author.

## AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual contribution to the work, and approved it for publication.

## FUNDING

This paper was partly funded by the DIH-HERO project from the European Union's Horizon 2020 Research and Innovation Programme under grant agreement no. 825003, and by the Ministry of Education, Science and Technological Development of the Republic of Serbia, under contracts (BL and KJ) TR-35003 and (TŠ) TR-33020.

## ACKNOWLEDGMENTS

Authors of this paper want to thank the Predrag Todorov for his contribution with the experiments and laboratory setup construction and to Prof. Stevica Graovac for sharing his equipment required for laboratory setup.

## SUPPLEMENTARY MATERIAL

The Supplementary Material for this article can be found online at: https://www.frontiersin.org/articles/10.3389/fnbot. 2019.00069/full#supplementary-material

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**Conflict of Interest Statement:** The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Copyright © 2019 Luki´c, Jovanovi´c and Šekara. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

# A System-of-Systems Bio-Inspired Design Process: Conceptual Design and Physical Prototype of a Reconfigurable Robot Capable of Multi-Modal Locomotion

Ning Tan<sup>1</sup> \*, Zhenglong Sun2,3 \*, Rajesh Elara Mohan<sup>4</sup> , Nishann Brahmananthan<sup>4</sup> , Srinivasan Venkataraman<sup>5</sup> , Ricardo Sosa<sup>6</sup> and Kristin Wood<sup>4</sup>

*<sup>1</sup> Key Laboratory of Machine Intelligence and Advanced Computing, Ministry of Education, School of Data and Computer Science, Sun Yat-sen University, Guangzhou, China, <sup>2</sup> School of Science and Engineering, Chinese University of Hong Kong, Shenzhen, China, <sup>3</sup> Shenzhen Institute of Artificial Intelligence and Robotics for Society, Shenzhen, China, <sup>4</sup> Engineering Products Development Pillar, Singapore University of Technology and Design, Singapore, Singapore, <sup>5</sup> Department of Design, Indian Institute of Technology Delhi, New Delhi, India, <sup>6</sup> Art Design & Architecture, Monash University, Melbourne, VIC, Australia*

#### Edited by: *Zhao Guo,*

*Wuhan University, China*

### Reviewed by:

*Zhibin Song, Key Laboratory of Mechanism Theory and Equipment Design of Ministry of Education, School of Mechanical Engineering, Tianjin University, China Valery E. Karpov, National Research University Higher School of Economics, Russia*

#### \*Correspondence:

*Ning Tan<EMAIL_ADDRESS>Zhenglong Sun<EMAIL_ADDRESS>
#### Received: *24 April 2019* Accepted: *04 September 2019* Published: *20 September 2019*

#### Citation:

*Tan N, Sun Z, Mohan RE, Brahmananthan N, Venkataraman S, Sosa R and Wood K (2019) A System-of-Systems Bio-Inspired Design Process: Conceptual Design and Physical Prototype of a Reconfigurable Robot Capable of Multi-Modal Locomotion. Front. Neurorobot. 13:78. doi: 10.3389/fnbot.2019.00078* Modern engineering problems require solutions with multiple functionalities in order to meet their practical needs to handle a variety of applications in different scenarios. Conventional design paradigms for single design purpose may not be able to satisfy this requirement efficiently. This paper proposes a novel system-of-systems bio-inspired design method framed in a solution-driven bio-inspired design paradigm. The whole design process consists of eight steps, that is, (1) biological solutions identification, (2) biological solutions definition/champion biological solutions, (3) principle extraction from each champion biological solution, (4) merging of extracted principles, (5) solution reframing, (6) problem search, (7) problem definition, and (8) principles application & implementation. The steps are elaborated and a case study of reconfigurable robots is presented following these eight steps. The design originates from the multimodal locomotion capabilities of two species (i.e., spiders and primates) and is analyzed based on the Pugh analysis. The resulting robotic platform could be potentially used for urban patrolling purposes.

Keywords: bio-inspired design, system-of-systems, multi-model locomotion, reconfigurable robots, mobile robotics

## INTRODUCTION

A design process is a systematic approach followed by designers while trying to solve a problem, which could be as simple as designing a chair or could be as complex as designing an aircraft (Mas et al., 2012). Irrespective of the complexity of the problem, looking for inspiration before starting to design has been a normative step in the world of design (Eckert and Stacey, 2000). Especially when it comes to exploration and discovery of new ideas (Murakami and Nakajima, 1997). Therefore, designers have started to follow a systematic approach to seek inspirations outside the problem domain to find solutions (i.e., domain-independent) where the problems are closely related to the original problem domain (López-Mesa, 2011).

The bio-inspired engineering design approach is one of systematic approaches using analogies from biological creatures in the nature to develop solutions for handling engineering problems (Helms et al., 2009; Eroglu et al., 2011a). There are many practical examples, such as the invention of Velcro (Versos and Coelho, 2011) and conceptual design of the Bionic Car project (Vincent and Man, 2002; Floyd et al., 2006). This also applies to the scientific world, for example, the nanoscale superhydrophobic coatings inspired by the self-cleaning mechanism of lotus leaves (Cheng and Rodak, 2005), the imitation of the pinecones to design clothes that can regulate body temperature (Groeneveld, 2008), and the design of microrobots that can walk on water, mimicking the locomotion of the basilisk lizard (Zari, 2007).

Modern engineering problems often require solutions with intrinsic compliance, for better function variety, environment adaptivity, and structure flexibility. As a result, soft materials start to emerge in the bio-inspired robotics design, such as the traditional cable-driven mechanism, spring-damper structure, and recent pneumatic artificial muscles. But the intrinsic drawback of using these materials is their lacking in the accuracy and reliability, making the design loss in robustness. As an alternative solution, reconfigurable design in robot becomes appealing, where not only it inherits the accuracy and robustness of rigid structure, but also it is able to achieve the compliance as desired using reconfiguration. In such a case, the intrinsic compliance roots in the reconfigurability, instead of the materials and flexible structures.

However, a systematic design methodology for such bioinspired design is still missing, making the process deducing challenging (French, 1994; Benyus, 1997). There are two main different approaches with respect to different starting points, that is, the Problem-Based Bio-Inspired Design (PB-BID) process and the Solution-Based Bio-Inspired Design (SB-BID) process (Eroglu et al., 2011b). In order to include a wide variety of functionalities at the beginning of the design process, here we stick to the SB-BID process since it is more appropriate to implement the system-of-systems paradigm.  In general, both Aalbog's and Helms' models begin with the identification of biological solutions, and then extract principles from the identified solution. The latter method involves the reframing and application of extracted principles to solve a real-world engineering problem. However, these previous models mainly led to creating a solution inspired by a single biological species for solving a single problem, with poor extensibility.

In this paper, we presented a novel BID process to creating engineering product with multiple functionalities. The inspiration is originated from multiple biological species to solve multiple problems. Since the process involves multiple biological systems (i.e., species), we name it as the System-of-Systems Bio-Inspired Design (SoS-BID) process. The process is firstly introduced and a case study will be demonstrated followed this design process, for designing a reconfigurable robot called Scorpio, which is able to change its morphologies to achieve different locomotion modes, namely crawling, rolling, and wall-climbing.

## SYSTEM-OF-SYSTEMS BIO-INSPIRED ENGINEERING DESIGN PROCESS

Typically, a design process would be dynamic, which changes frequently based on the context of the problem faced and the available solution space (Vattam et al., 2007). Thus, the process undergoes repetitive reformulations of both design problems and solutions. In the following sections, we present the proposed SoS-BID process to solve the design problem of interest. The process is a systematic approach, which includes the following eight steps:

A systematic diagram, explaining an overview of the proposed SoS-BID process framework is shown in **Figure 1**.

## Step 1: Biological Solutions Identification

In the very first step of the SoS-BID process, designers start with multiple biological solutions of interest as inspiration sources that will potentially be used to solve a certain problem (Lindemann and Gramann, 2004). This step involves the search of biological species which perform desired biologized tasks. Instead of randomly observing, here we propose to observe the biological species based on the taxonomy which is the science of defining biological species based on shared characteristics. The list of species could be categorized according to their specific abilities or nature (Huxley, 1875; Shane et al., 1986; Santori et al., 2005; Alexander, 2006; Manohar et al., 2016) as follows:

Once the categories are defined, the biological solution search is conducted under each specifically defined category. The output of this step would be a series of categories and a list of biological species for each category.

## Step 2: Biological Solutions Definition/Champion Biological Solutions

It is noted that the selection from the biological solutions identified in Step 1 could be subjective. However, we have defined a few general selection criteria of the species to assist the designers in decision making, such as the task performance efficiency, multi-functional capability, ease of principle extraction, ease of kinematic study, and practical feasibility. At this step, the following existing design-concept-selection methods can be adopted for analysis:

The selected species is called the champion biological solution/species. The output of this step would result in a single champion biological species selected for each category.

## Step 3: Principle Extraction From Each Champion Biological Solution

This step involves a deeper understanding of the selected biological species regarding their functions and behavior, and identification of the underlying principles used to solve a problem in the context of nature. Finally, through the analysis of the species, the important fundamental principles are extracted from the champion biological species.

## Step 4: Merging of Extracted Principles

The extracted principles are analyzed for identifying repetitive principles. This step involves the removal of repetitive principles and merging of the resulting principles into a series of unique abstract principles (functions, behaviors, etc.) in general.

## Step 5: Solution Reframing

This step is to transfer the solution from the biological domain into the engineering domain. In particular, the reframing step forces the designers to think how humans might view the significance of the biological principles extracted. The output of this process would result in a series of useful applications and related constraints of the biological principles in the humansociety context, which are labeled as "humanized solutions."

## Step 6: Problem Search

Given the generated humanized solution and related constraints, this step is to find a potential engineering problem that the solution could be applied to. The problem could be an existing one or an entirely newly defined problem.

## Step 7: Problem Definition

The problem definition plays an important role in the SoS-BID process, involving deep and higher level understanding and interpretation of the searched problem. Generally, this step includes the following sub-steps:

It is worth noting that the resultant criteria generated above may still be abstract, based on which the specific and detailed system requirements are to be derived.

## Step 8: Principles Application & Implementation

This step involves the real transformation and implementation of the principles identified in the biological domain into the engineering domain. In other words, the biological-solutionapplicable principles have been translated into engineering terminologies to facilitate the process of implementation. The output of this step results in an embodied principle of the engineering concept that satisfies the need of one or several practical engineering problems as defined in the previous step.

To capture the general features of the SoS-BID process, a summary of the full process is presented in **Table 1** with some short descriptions.

TABLE 1 | A summary of the system-of-systems bio-inspired design process.

## DEVELOPMENT OF THE SCORPIO ROBOT USING THE PROPOSED SYSTEM-OF-SYSTEMS BIO-INSPIRED DESIGN PROCESS

## Step 1: Biological Solutions Identification

To begin the SoS BID process, we decide to look at the species with shared characteristics of their locomotion capabilities. The locomotion aspect of biological species refers to that how biological species in nature maneuver from one point to another. Here the locomotion modes are categorized into two major categories of interests:

In such manner, the locomotion in space can be seen as the combination of these two types of locomotion. Based on the two categories, the biological species are identified using academic articles from Google scholar and non-academic articles from other online search engines, and categorized as follows.

## Category 1: Species Which Can Perform Planar Locomotion

It is found that most of the species are capable of performing multiple locomotion modes (Jenkins, 2012; Lock et al., 2013). Especially, they perform different locomotion modes based on the encountered different scenarios (Kuroda et al., 2014). Some are even capable of changing their shapes in order to perform multiple locomotion gaits (Prostak, 2014; Nemoto et al., 2015; Mintchev and Floreano, 2016; Grzimek's Animal Life Encyclopedia, 2017). Species in nature such as snakes, lizards, and insects can adapt their gaits in response to different types of terrains that they navigate, which vary from smooth terrains to bumpy ones (Bagheri et al., 2015). Most species switch their locomotion modes to traverse different terrains, for example, to overcome obstacles such as pits and bumpy surfaces on their ways. Hereby, we select five species in nature, which can change their shape to perform multiple locomotion modes, for further analysis.

• Mount Lyell Salamander

Salamanders generate tetrapod postures to help them to walk (Cabelguen et al., 2010). A larger portion of its energy is used up for lifting their body for walking which results in a slower motion. Most salamanders found in nature can partially curl up their body, tails, and legs as a defense mechanism and limits them from performing rolling locomotion. The Mount Lyell salamander is a species found in the northern Sierra Nevada in California, which can curl up its entire body to form a spherical shape (Mintchev and Floreano, 2016). Such a spherical morphology enables it to roll down a slope without getting injured (King, 2013).

• Woodlouse

Woodlouse is an isopod species which belongs to the Armadillidae family which normally walks most of the time. It is also capable of rolling up its entire body to form a spherical shape as a defense mechanism and to roll down a slope without getting injured (Grzimek's Animal Life Encyclopedia, 2017).

• Moth Caterpillar

Moth caterpillar belongs to the Lepidoptera family which is also capable of walking and rolling. While being poked, it will perform a backward roll by curling up into a ball shape. The curling movement triggers a momentum for rolling locomotion. Once it depleted the momentum, it needs to relax and trigger a new curling movement to perform the rolling locomotion once again (King, 2013).

• Wheel Spider

The wheel spider is a kind of huntsman spider found in Namib desert. The spider possesses two types of locomotion modality, i.e., crawling and rolling (Nemoto et al., 2015). While entering a slope, it flips sideways and runs cartwheel down the slope, which can produce around 44 turns per second (Leroy and Leroy, 2003).

• Cebrennus rechenbergi

Cebrennus rechenbergi is found in Morocco deserts, belongs to the family of huntsman spider, which is also known as the Moroccan flicflac spider. Like the species mentioned above, such a kind of spider is also capable of two types of locomotion modality, namely crawling and rolling. It is also known uniquely for its rolling locomotion within the spiders' family. Once being threatened, it can multiply its speed by performing the acrobatic flicflac somersault locomotion (**Figure 2**) which assists to propel off the ground to go uphill, downhill, and on flat terrain (Prostak, 2014).

**Table 2** showcases the different locomotion modes performed by the species in Category 1.

TABLE 2 | Locomotion performed by five species of Category 1.

## Category 2: Species Which Can Perform Vertical Locomotion